Foundations of mathematics. 2. Set theory (continued)


 Beverly Horton
 3 years ago
 Views:
Transcription
1 2.1. Tuples, amilies Foundations o mathematics 2. Set theory (continued) Sylvain Poirier settheory.net A tuple (or ntuple, or any integer n) is an interpretation o a list o n variables. It is thus a metaunction rom a inite metaset, to the universe. Tuples o a given kind (list o variables with their types) can be added to any theory as a new type o objects, whose variables are meant as abbreviations o packs o n variables (copies o the list with the same old types) x = (x 0,, x n 1 ). In practice, the domain o considered ntuples will be the (meta)set V n o n digits rom 0 to n 1. Set theory can represent its own ntuples as unctions, iguring V n as a set o objects all named by constants. A 2tuple is called an oriented pair, a 3tuple is a triple, a 4tuple is a quadruple... The ntuple deiner is not a binder but an nary operator, placing its n arguments in a parenthesis and separated by commas: (,, ). The evaluator appears (curried by ixing the metaargument) as a list o n unctors called projections: or each i V n, the ith projection π i gives the value π i (x) = x i o each tuple x = (x 0,, x n 1 ) at i (value o the ith variable inside x). They are subject to the ollowing axioms (where the irst sums up the next ones) : or any x 0,, x n 1 and any ntuple x, x = (x 0,, x n 1 ) (π 0 (x) = x 0 π n 1 (x) = x n 1 ) x i = π i ((x 0,, x n 1 )) x = (π 0 (x),, π n 1 (x)) or each i V n Oriented pairs suice to build (copies o) ntuples or each n > 2, in the sense that we can deine operators to play the roles o deiner and projections, satisying the same axioms. For example, triples t = (x, y, z) can be deined as t = ((x, y), z)) and evaluated by x = π 0 (π 0 (t)), y = π 1 (π 0 (t)), z = π 1 (t). Conditional connective The 3ary connective I A then B else C, is written as ollows (applying (C, B) to A V 2 ) : (A B C) ( C A B) ((A B) ( A C)) ( A C B) (C, B)(A) ((A B) ( A C)) ((C A) (A B)) (A B C) Any n+1ary connective K amounts to two nary ones: K(A) (A K(1) K(0)). Thus, A (A 0 1); (A B) (A B 1); (A B) (A 1 B); (A B) (A B B). Families A amily is a unction intuitively seen as a generalized tuple: its domain (a set) is seen as simple, ixed, outside the main studied system, as i it was a set o metaobjects. A amily o... is a amily whose image is a set o.... Families use the ormalism o unctions disguised in the style o tuples (whose tools cannot apply due to the initeness o symbols). So the notation u i that looks like a metavariable symbol o variable, evaluates u at i (abbreviating u(i) or π i (u)). A amily deined by a term t, is written (t(i)) instead o (I i t(i)) or (t(0),, t(n 1)). The argument i is called index, and the amily is said indexed by its domain I. A amily indexed by the set N o integers is called a sequence. Structures and binding symbols Each nary structure can be interpreted as a unary structure whose class o deiniteness is a class o ntuples, just like a binder can be seen as a unary structure over a class o unctions or subsets rom the given set. Indeed, binders are the generalization o structures when tuples are replaced by amilies. In particular, quantiiers and are the respective generalizations o chains o conjunctions and disjunctions: (B 0 B n 1 ) ( i V n, B i ). We can see the equality condition between ordered pairs, (x, y) = (z, t) (x = z y = t), as similar to the one or unctions. Let R a unary predicate deinite on E, and C a boolean. We have distributivities (C x E, R(x)) ( x E, C R(x)) (C x E, R(x)) ( x E, C R(x)) (C x E, R(x)) ( x E, C R(x)) (( x E, R(x)) C) ( x E, R(x) C) ( x E, C) (C E ) C (C E = ) ( x E, C) 1
2 Extensional deinitions o sets The unctor Im deines the binder {T (x) x E} = {T (x)} x E = Im(E x T (x)). As this notation looks similar to the set builder, we can combine both : {T (x) x E R(x)} = {T (x) x {y E R(y)}} Applying Im to tuples (a, b, ), deines the operator symbols o exensional deinition o sets (listing their elements), written {a, b, }. For example V n = {0,, n 1}. Those o arity 0,1,2 are the empty set, the singleton {a} and the pairing {a, b} already presented in Images o tuples are inite sets (initeness will be ormally deined in text 5, with Galois connections) Boolean operators on amilies o sets Union o a amily o sets For any list o unary predicates A i with index i I, all deinite in (at least) a common class C, applying them to the same variable x with range C reduces them to Booleans whose value depends in x i.e. Boolean variables with parameter x (in the sense o 1.5). This way, any Boolean operation between these variables (a connector or quantiier Q) deines a metaoperation between unary predicates, with result the unary predicate R deined by C x, (R(x) Qi, A i (x)). When C is a set E, this operates between subsets o E (through and the set builder). For example, the quantiier Q = deines the union o a amily o sets: x F i i I, x F i This class is a set independent o E, as it can also be deined rom the union o a set o sets (1.11): F i = {F i i I} ( x F i, B(x)) i I, x F i, B(x) x A B (x A x B) A A B = B A All extensional deinition operators (except ) are deinable rom pairing and binary union. Intersection Now any ixed amily o sets (F i ) can be seen as a amily o subsets o some common set, such as their union U = F i, or any other set E such that U E. Then or any operation (quantiier) Q between Booleans indexed by I, the predicate R(x) deined as (Qi, x F i ) takes value (Qi, 0) or all x / U. Thus Q needs to satisy (Qi, 0) (i.e. to be alse when all entries are alse) or the class R to be a set, then expressible as {x E R(x)}, or as {x U R(x)}. This condition was always satisied or the quantiier (that deines the union) including on the empty amily ( = ), but or (that deines the intersection) it requires a nonempty amily o sets (i.e. (F i ) with I ): j I, F i = {x F j i I, x F i } x F i i I, x F i x A B (x A x B) A = A B B A B = A B A B = B A = {x A x B} A Two sets A and B are called disjoint when A B =, which is equivalent to x A, x / B. Union and intersection have the same associativity and distributivity properties as and : A B C = (A B) C = A (B C) = {A, B, C} ( A i ) C = i C) (A ( A i ) C = i C) (A (A B) C = (A C) (B C) (A B) C = (A C) (B C) Other operators The dierence is deined by A\B = {x A x / B} so that x A\B x A x / B. Finally the connector gives the symmetric dierence: A B = (A B)\(A B). When (Qi, 0) is true, we must choose a set E to deine operations between subsets o E : Negation deines E F = E\F, called the complement o F in E : x E, x F x E F. The intersection o the empty amily gives E: or a amily o subsets F i o E, F i = {F i } = {x E i I, x F i } = E E F i 2
3 2.3. Products, graphs and composition Finite product For two sets E and F, the product E F is the set o (x, y) where x E and y F. Similarly, the product o n sets E 0 E n 1 is the set o ntuples (x 0,, x n 1 ) where i V n, x i E i. An nary operation is a unction with domain a product o n sets. A relation (or example between E and F ) can be expressed as a set o tuples (G E F ). The domains E and F can be speciied by taking the triple (E, F, G). A set o oriented pairs (such as G) is called a graph. For any binder L and any graph G, the ormula Lx G, S(x 0, x 1 ) that binds x = (x 0, x 1 ) on a binary structure S with domain G, can be seen as binding 2 variables x 0, x 1 on S(x 0, x 1 ), and thus be denoted with an oriented pair o variables: L(y, z) G, S(y, z). The existence o the product (in all arity) is justiied by the set generation principle: ( (x, y) E F, R(x, y)) ( x E, y F, R(x, y)) ( y F, x E, R(x, y)) ( (x, y) E F, R(x, y)) ( x E, y F, R(x, y)) ( y F, x E, R(x, y)) The quantiier (x, y) E E will be abbreviated as x, y E, and the same or. When F = V 2, ( x E, A(x) B(x)) (( x E, A(x)) ( x E, B(x)) ( x E, A(x) B(x)) (( x E, A(x)) ( x E, B(x)) ( x E, C A(x)) ((C (E )) x E, A(x)) ( x E, C A(x)) ((C (E = )) x E, A(x)) Sum or disjoint union The transpose o an ordered pair is t (x, y) = (y, x); that o a graph R is t R = {(y, x) (x, y) R}. Graphs can be expressed in curried orms R and R : y R(x) (x, y) R x R(y) = t R(y), justiied by deining the unctor R as R(x) = {y (z, y) R (z = x)}. Inversely, any amily o sets (E i ) I I deines a graph called their sum (or disjoint union) E i: (i I x E i ) (i, x) E i = {i} E i = {(i, x) x E i } I E i ( x E i, A(x)) ( i I, y E i, A(i, y)) E 0 E n 1 = i V n E i E F = x E F E = = E (E E F F ) E F E F ( i I, E i E i) E i E i Composition, restriction, graph o a unction For any set E, the unction identity on E is deined by Id E = (E x x). For any unctions, g with Im Dom g (namely, : E F and g : F G), their composite is g = ((Dom ) x g((x))) : E G The same with h : G H, h g = (h g) = h (g ) = (E x h(g((x)))) and so on. The restriction o to A Dom is A = (A x (x)) = Id A. The graph o a unction is deined by Gr = {(x, (x)) x Dom } = x Dom {(x)} (x, y) Gr (x Dom y = (x)) We can deine domains, images and restrictions or graphs, letting those or unctions be particular cases (i.e. Dom = Dom(Gr ), Im = Im(Gr ) and Gr( A ) = (Gr ) A ): Dom R = {x (x, y) R} = Im t R x, x Dom R R(x) R E F (Dom R E Im R F ) R A = R (A Im R) = {(x, y) R x A} = x A R(x) 3
4 Then we have R = E i (Dom R I i I, R(i) = E i ) Im R = E i. For any unctions, g, any graph R, and E = Dom, Gr R x E, (x) R(x) R Gr ( (x, y) R, x E y = (x)) (Dom R E (x, y) R, y = (x)) Gr Gr g ((E Dom g) = g E ) Direct image, inverse image The direct image o a set A by a graph R is R (A) = Im R A = x A R(x). Dom R A R A = R R (A) = Im R R ( A i ) = R (A i ) R ( A i ) R (A i ) A B R (A) R (B) The direct image o A Dom by a unction is [A] = (A) = (Gr ) (A) = Im( A ) = {(x) x A} Im For any : E F and B F, the inverse image o B by, written (B), is deined by (B) = ( t Gr ) (B) = {x E (x) B} = (y) y B (y) = ( Gr)(y) = ({y}) = {x E (x) = y} ( F B) = E (B) For any amily (B i ) o subsets o F, ( B i) = (B i ) where intersections are respectively interpreted as subsets o F and E Uniqueness quantiiers, unctional graphs For all sets F E, all unary predicate A deinite on E, and all x E, x F {x} F ( y E, x = y y F ) ( y E, x = y y F ) x F (( y F, A(y)) A(x) y F, A(y)) F {x} ( y F, x = y) (( y F, A(y)) A(x) ( y F, A(y))) F = {x} (x F y F, x = y) ( y E, y F x = y) Here are 3 new quantiiers: 2 (plurality),! (uniqueness), and! (existence and uniqueness), whose results when applied to R in E only depend on F = {x E, R(x)} (like and unlike ) : ( x E, R(x)) (F ) ( x F, 1) ( x E, {x} F ) ( 2x E, R(x)) ( 2 : F ) ( x, y F, x y) ( x, y E, R(x) R(y) x y) (!x E, R(x)) (!: F ) ( 2 : F ) ( x, y F, x = y) x F, F {x} (!x E, R(x)) (! : F ) ( x F, F {x}) ( x E, F = {x}) F {x} y F, F {y} (!: F ) (! : F ) (F!: F ) F (( y F, A(y)) ( y F, A(y))) (!: F ) (( y F, A(y)) ( y F, A(y))) F = {x} (( y F, A(y)) A(x) y F, A(y)) A unction is said constant when!: Im. The constancy o a tuple is the chain o equalities: x = y = z!:{x, y, z} ((x = y) (y = z)) x = z 4
5 Translating operators into predicates In a generic theory, any unctor symbol T can be replaced by a binary predicate symbol R (where x R y (y = T (x))) with the axiom x,!y, x R y, replacing any ormula A(T (x)) (where x is a term) by ( y, x R y A(y)), or by ( y, x R y A(y)) (while terms cannot be translated). This way, any predicate R such that x,!y, x R y implicitly deines an operator symbol T. We can extend this to other arities by replacing x by a tuple. But the use o open quantiiers in this construction makes it unacceptable in our set theory. Instead, let us introduce a new operator ɛ on the class (Set(E)! : E) o singletons, giving their element according to the axiom ( x)ɛ{x} = x, or equivalently (Set E! : E) ɛe E. Then or every unary predicate A and every singleton E, A(ɛE) ( x E, A(x)) ( x E, A(x)). Conditional operator Like the conditional connector, it chooses between two objects x, y depending on the boolean B: (B x y) = (y, x) B = ɛ{z {x, y} B z = x z = y} so that or any predicate A we have A(B x y) (B A(x) A(y)). Combined with structures, it is the natural means to deine any other structuring paraoperator (like the one translating booleans into objects), making useless their direct registration to the language o a theory. Functional graphs A graph R is said unctional i x Dom R,!: R(x), or equivalently x, y R, x 0 = y 0 x 1 = y 1. This is the condition or it to be the graph o a unction. Namely, R = Gr(ɛ R) where 2.5. The powerset axiom ɛ R = ((Dom R) x ɛ( R(x))) Let us extend set theory by 3 new symbols (powerset, exponentiation, product) together with axioms, that will declare given classes C to be sets K. Such extensions are similar to those given by the set generation principle (1.11), except that they no more satisy its condition o application. In the traditional ZF ormalization o set theory only accepting as primitive stucture, such a declaration is done purely by the axiom (or theorem) ( parameters), K, x, x K C(x). Then, K can be used to represent the ollowing abbreviations: x K means x, C(x) ; the equality X = K means ( x, x X C(x)), and any other A(K) means X, (X = K) A(X). But these ormulas use open quantiiers, orbidden in our ramework. Cases justiied by the set generation principle could still be processed by replacing C by ormulas, while the above X could be interpreted by the axiom with existential elimination (1.9). But this no more works in other cases : without open quantiiers, even a given set K that would happen to be identical to a class C, would still not be recognizable as such, so that the very claim that C coincides with a set remains practically meaningless. Thus instead o this, the eective ormalization requires to name K by a new primitive symbol o set theory, with argument the tuple y o parameters o C, with the axiom ( y ), Set(K(y)) x, x K(y) C y (x) Powerset. Set E, Set(P(E)) F, F P(E) (Set(F ) F E). We shall also shorten P into in binding symbols: ( A E, ) ( A P(E), ). Cantor Theorem. ( Set E, Fnc ) Dom = E P(E) Im. Proo: F = {x E x / (x)} ( x E, x F x (x)) ( x E, F (x)) F / Im. (The Russell paradox may be seen as a particular case) Exponentiation. Set E, F, Set(F E ), ( F E : E F ). Product o a amily o sets. This binder is the generalization o inite product operators : x, x E i (Fnc(x) Dom x = I i I, x i E i ). For all i I we call ith projection, the unction π i rom j I E j to E i evaluating every amily x at i : π i (x) = x i. This is the unction evaluator seen as curried in the unusual order. 5
6 These operators are equivalent in the sense that they are deinable rom each other: P(E) = {{x E (x) = 1} V 2 E } F E = F = {ɛ R R E F x E,! : R(x)} x E E i x E,! : R(x)} E i = {x ( E i ) I i I, x i E i } = {ɛ R R Even some cases are expressible rom previous tools: F {a} = {{a} x y y F } F = { } P({a}) = {, {a}} { E i = (i I x i y i ) J (x, y) E i } E i J i J ( i I,E i = ) E i = ( i I,! : E i ) E i = {(ɛe i ) } I F F then P(F ) P(F ), F E F E, and ( i I, E i E i) 2.6. Injectivity and inversion E i E i. A unction : E F is injective (or : an injection) i t Gr is a unctional graph: Inj ( y F,!: (y)) ( x, x E, (x) = (x ) x = x ) ( x, x E, x x (x) (x )) Then its inverse is deined by 1 = ɛ = (Im y ɛ (y)) so that Gr( 1 ) = t Gr. A unction : E F is said bijective (or a bijection) rom E onto F, and we write : E F, i it is injective and surjective: y F,! : (y), in which case 1 : F E. A permutation (or transormation) o a set E is a bijection : E E. Proposition. Let : E F and g : F G. 1. (Inj Inj g) Inj(g ) 2. Inj(g ) Inj 3. Im(g ) = g[im ] Im g. 4. Im(g ) = G Im g = G. 5. Im = F Im(g ) = Im g, so that (( : E F ) (g : F G)) (g : E G) Proos: 1. (Inj Inj g) x, y E, g((x)) = g((y)) (x) = (y) x = y. 2. x, y E, (x) = (y) g((x)) = (g((y)) x = y. 3. z G, z Im(g ) ( x E, g((x)) = z) ( y Im, g(y) = z) z g[im ] and are obvious. Proposition. For any sets E, F, G and any F E, Im = F Inj(G F g g ) (Im = F!: G) (Inj G ) {g g G F } = G E (Inj!: G) (E F G ) {g E g G F } = G E (Inj E ) g E F, g = Id E Proos : Im = F g, h G F, (( x E, g((x)) = h((x))) ( y F, g(y) = h(y) g = h)). z, z G, (y z) = (y (y Im z z )) thus Inj(g g ) (z = z Im = F ). z G, h G E, Inj (F y (y Im h 1 (y) z)) = h. z, z G, x E, i g G F is such that y E, g((y)) = (y = x z z ), then y E, (y) = (x) g((y)) = g((x)) = z (y = x z = z ). The last ormulas are particular cases o the second one. Let : E F and F : P(F ) P(E) deined by. Then (by G = V 2 and B Im, [ (B)] = B), Inj Im F = P(E) Im = F Inj F Im = {[A] A E} = P(Im ) 6
7 Proposition. Let F = Dom g. Then Inj g Inj(F E g ) (Inj g E = ). Proos:, F E, (Inj g x E, g((x)) = g( (x))) ( x E, (x) = (x)). Then rom the middle ormula, y, z F, g(y) = g(z) g (E x y) = g (E x z) ( x E, y = z) (y = z E = ). Proposition. Let : E F and Dom g = F. Then g = Id E ( x E, y F, (x) = y g(y) = x) (Gr t Gr g) ( y F, (y) {g(y)}) ( x E, (x) g (x)) (Inj g Im = 1 ) E Im g (g : F E g = Id E ) (Im = F Inj g g = Id F g = 1 ). Proo o the last ormula: (Inj g g g = g Id F ) g = Id F. Proposition. 1) I, g are injective and Im = Dom g, then (g ) 1 = 1 g 1. 2) I, h : E F and g : F E then (g = Id E h g = Id F ) ((g : F E) = h = g 1 ). Proos: 1) x Dom, y Im g, (g )(x) = y (x) = g 1 (y) x = ( 1 g 1 )(y). Other method: (g ) 1 = (g ) 1 g g 1 = (g ) 1 g 1 g 1 = 1 g 1. 2) We deduce = h rom = h g = h, or rom Gr t Gr g Gr h. The rest is obvious Properties o binary relations on a set ; ordered sets A binary relation on E is a graph R E E. Denoting x R y (x, y) R and omitting the domain E o quantiiers, it will be said relexive x, x R x irrelexive x, (x R x) symmetric R t R R = t R R = R antisymmetric x, y, (x R y y R x) x = y transitive x, y, z, (x R y y R z) x R z For any transitive binary relation R we denote x R y R z ((x R y) (y R z)) x R z. Example. Let A E E and R = A Gr. Then (Id E A) R is relexive (, g A, g A) R is transitive ( A, : E E 1 A) R is symmetric Preorder. A preorder R on a set E is a relexive and transitive binary relation on E. Equivalently, x, y E, x R y R(x) R(y) Proo: x, y E, x R(x) R(y) x R y ; transitivity says x R y R(x) R(y). Note : t R is then also a preorder, i.e. x R y R(y) R(x). Ordered set. An order is an antisymmetric preorder. A preordered set is a set E together with a preorder R (thus an ordered pair (E, R)). An ordered set is a set with an order, usually written as. For x, y in an ordered set E, the ormula x y can be read x is less than y, or y is greater than x. The elements x and y are said incomparable when (x y y x). (This implies x y). Any subset F o a set E with an order (resp. a preorder) R E E, is also ordered (resp. preordered) by its restriction R (F F ) (which is an order, resp. preorder, on F ). Strict order. It is a binary relation both transitive and irrelexive; and thus also antisymmetric. Strict orders < bijectively correspond to orders by x < y (x y x y). The inverse correspondence is deined by x y (x < y x = y). Total order. A total order on a set E is an order R on E such that R t R = E E. Namely, it is an order where no two elements are incomparable. Equivalently, it is an order related with its strict order < by x, y E, x < y y x. Still equivalently, it is a transitive relation such that x, y E, x y (y x x = y). A strict order associated with a total order, called a strict total order, is any transitive relation < in E such that x, y E, x < y (y < x x = y), or equivalently x, y E, x = y (y < x x < y)). 7
8 Monotone, antitone, strictly monotone unctions Between ordered sets E and F, a unction : E F will be said : monotone i x, y E, x y (x) (y) antitone i x, y E, x y (y) (x) strictly monotone i x, y E, x y (x) (y) strictly antitone i x, y E, x y (y) (x). Any composite o a chain o monotone or antitone unctions, is monotone i the number o antitone unctions in the chain is even, or antitone i it is odd. Any strictly monotone or strictly antitone unction is injective. I F E and g E F are both monotone (resp. both antitone) and g = Id E, then is strictly monotone (resp. strictly antitone). In each ordered set E, a unction E E is said extensive i x E, x (x), i.e. Id E. The composite o two extensive unctions is extensive. Order on sets o unctions For all sets E, F where F is ordered, the set F E (and thus any subset o F E ) is ordered by g ( x E, (x) g(x)) Then,, g F E, h E G, g h g h, i.e. h is always monotone. The particular case F = V 2 is that P(E) (and thus any set o sets) is naturally ordered by, and that h is monotone rom P(E) to P(G). I F and G are ordered and u G F is monotone (resp. antitone) then F E u G E is monotone (resp. antitone) Canonical bijections For all objects x, y we shall say that x determines y i there is an invariant unctor T such that T (x) = y. Then the role o y as ree variable can be played by the term T (x), thus by the use o x. This is a preorder on the universe, but not a predicate o set theory as it involves a metaconcept. Instead, it is meant to abbreviate the use o T. Similarly, a unction : E F will be said canonical i it is deined as E x T (x) or some invariant unctor T. A bijection will be said bicanonical i both and 1 are canonical. When a bijection : E F is canonical (resp. bicanonical), we write : E = F (resp. : E F ); or, using its deining unctor, T : E = F which means that (E x T (x)) is injective with image F. We shall write E = F (resp. E F ) to mean the existence o a canonical (resp. bicanonical) bijection, that is kept implicit. Canonical bijections can ail to be bicanonical especially when their deining unctor is not injective: V 2 E = P(E), {x} E = {x} and E {x} = E, whereas {x} E E {x} and E {0} E. This is a preorder on the class o sets, preserved by constructions: or example i E = E and F = F then E F = E F and F E = F E using the direct image o the graph (while we may not have F E = F E when E does not determine E). It will oten look like a property o numbers as the existence o a bijection between inite sets implies the equality o their numbers o elements. The transposition o oriented pairs (E F F E) extends to graphs (P(E F ) P(F E)) and to operations: G E F G F E where G E F is transposed by t (x, y) = (y, x). Sums o sets, sums o unctions I S = E i then : P(E i) = P(S), whose inverse S R R I = (I i R(i)) depends on I. In particular or two sets E and F we have (P(F )) E = P(E F ). The sum over I o unctions i where i I, E i = Dom i is deined by i = (S (i, x) i (x)) = i (Dom = S i I, i = (i) = j i ) where j i = (E i x (i, x)) Thus the canonical bijections (bicanonical i I = Dom S, thus i E in the case I E) F Ei = F S F (i,x) = F c x E i (F E ) I = F I E ( i ) (F E ) I c S (E F ) G E F G Gr i P(I (E F )) P((I E) F ) Gr i 8
9 Product o unctions or recurrying Transposing a relation R exchanges its curried orms R and R, by a bijection (P(F )) E (P(E)) F with parameter F. Similarly we have a bijection (F E ) I (F I ) E, canonical i I (to let (F E ) I determine E), deined by a binder called the product o the unctions i F E or i I: ( i ) F E i = ( F i ) E i = (E x ( i (x)) ), = i ( : E Im i i I, i = π i ) Dom = Dom g = E g = (E x ((x), g(x))) I E F E (I F ) E F i ) E. φ I E x E F φ(x) ( 2.9. Equivalence relations and partitions Indexed partitions A amily o sets (A i ) is called pairwise disjoint when any pair o them is disjoint : i, j I, i j A i A j = Equivalently, ( (i, x), (j, y) k I A k, x = y i = j), thus, Gr = t A i with Dom = Im A i = A i Im = {i I A i } Inversely, any F E deines a amily F = ( (y)) y F P(E) F o pairwise disjoint sets : y, z F, (y) (z) x (y) (z), y = (x) = z An indexed partition o a set E is a amily o nonempty, pairwise disjoint subsets o E, whose union is E. It is always injective : i, j I, A i = A j A i A j = A i i = j. Equivalence relation associated with a unction An equivalence relation is a symmetric preorder. Any : E F deines an equivalence relation on E by = {(x, y) E (x) = (y)} = ( ) Its composite g = h with any h G F satisies g, with = g Inj h Im = (h Im ) 1 g In particular, coincides with the equality relation Gr Id E on E when is injective. As =, the injectivity o the indexed partition Im (that we will abusively denote as ) gives the characteristic identity o equivalence relations : x y (x) = (y). When F = Im, this map G F h h {g G E g } is not only injective (by 2.6), but bijective. Indeed, with H = Im( g), ( g (y, z), (y, z ) H, y = y z = z ) Dom H = F ( h G F, g = h H Gr h). For any unctions, g such that Dom = Dom g, the unction with graph Im( g) is g called the quotient g/ : Im Im g, and is the only unction h such that Dom h = Im g = h. Inversion comes as a particular case: Inj 1 = Id Dom /. Remark. i R is relexive and x, y, z, (x R y z R y) z R x then R is an equivalence relation. Proo : symmetry is veriied as: x, y, (x R y y R y) y R x. Then comes transitivity. 9
10 Partition, canonical surjection A partition o E is a set o nonempty, pairwise disjoint sets whose union is E, thus the image o any indexed partition o E (where is any unction with domain E): P = Im = Im( ). For any binary relation R on E, i P = Im R E P(E) then ( x, y E, x R(y) R(x) = R(y)) ( x E, A P, x A R(x) = A) Id P = R Thus i R is an equivalence relation then P is a partition. Conversely or any partition P o E, the indexed partition Id P o E is = g or a unique g = R E P E. Then, P = Dom g = Im g thus R is an equivalence relation (deinable by x R y ( A P, x A y A)). The partition Im R associated with an equivalence relation R on E is called the quotient o E by R and denoted E/R; and the unction R is called canonical surjection rom E to E/R. For all x E, the element R(x), only element de E/R containing x, is called the class o x by R. Order quotient o a preorder Any preordered set (E, R) is relected through R by the ordered set (Im R, ), with R(x) = R(y) ( R(x) R(y) R(y) R(x)) (x R y y R x) so that R is injective i and only i R is an order; and the relation in Im R is a copy o the order quotient o the preorder R on the set E/(R t R). On each (ordered) set E, only one order will usually be considered, denoted E, or abusively. This may be justiied by deining ordered sets as sets o sets, ordered by, ignoring their elements Axiom o choice Let us review dierent equivalent orms o this claim, that is called an axiom as it cannot be deduced rom other axioms o set theory but it eels true and can conveniently be added as an axiom (that does not increase the risk o contractions but leads to some interesting theorems) ; but we will actually not need to do so in the rest o this work. Axiom o choice (AC). It says Set X, AC X, where AC X names the ollowing equivalent claims (1) Any product over X o nonempty sets is nonempty : ( x X, A x ) x X A x. (2) Set E, R X E, ( x X, y E, xry) ( E X, x X, xr(x)). Or in short : or any graph R, X = Dom R (Im R) X, Gr R. (3) Fnc g, Im g = X (Dom g) X, g = Id X. Proo o equivalence : (2) (1) by R = x X A x ; (1) (3) by A x = g (x) ; (3) (2) Im π 0 R = Dom R = X (h ) R X, h = Id X Gr R. (We also have (1) (2) by A x = R(x), and (2) (3) by R = t Gr g) Theorem. Each o the ollowing claims is equivalent to the axiom o choice: (4) For any set E o sets, / E ( A E A). (5) For any partition P o a set E, K E, A P,! : K A (6) For any sets E, F, G and any g : F G, {g F E } = G E. Proos: (1) (4) is obvious ; (4) (5) (x A P A K = Im x) (K E A P, x A A B P, x B A B A = B) (5) (3) Let P = Im g. Then = (X x ɛ(k g (x))) = g 1 K g = Id X. (AC E 2) (6) h G E, ( x E, y F, g(y) = h(x)) ( F E, x E, g((x)) = h(x)) (AC G 3) (6) : i F G, g i = Id G h G E, i h F E g i h = h. (6) (3) : E = G Id E {g F E }. Remarks: (4) (1) is also easy : / {A i i I} = E, then A E A ((A i)) A i. (6) has a converse : (Dom g = F E {g F E } = G E ) Im g = G. For explicitly inite X such as X = V n, a direct proo o AC X can be written with one variable per element o X. A general proo o the more abstract (X is inite AC X ) will be seen in text 5. 10
11 2.11. Notion o Galois connection The set o ixed points o a unction is written Fix = {x Dom (x) = x} Im. Then, Im Fix g ((Im Dom g) (g = )) Im = Fix ((Im Dom ) ( = )) : such a unction is called idempotent. Deinition. For any ordered sets E, F, and F = F with the transposed order, the sets o antitone Galois connections between E and F, and monotone Galois connections rom E to F, are deined by Gal(E, F ) = {(, ) F E E F x E, y F, x E (y) y F (x)} = t Gal(F, E) Gal + (E, F ) = {(u, v) F E E F x E, y F, x E v(y) u(x) F y} = Gal(E, F ) Fundamental example. Any relation R X Y deines a (, ) Gal(P(X), P(Y )) by A X, (A) = {y Y x A, x R y} = R(x) x A B Y, (B) = {x X y B, x R y} = {x X B R(x)} ( ) = Y ( ) = X Proo : A X, B F, A (B) ( x A, y B, x R y) B (A). This will later be shown to be a bijection : Gal(P(X), P(Y )) = P(X Y ). Lemma. F E,! E F, (, ) Gal(E, F ). Proo: E F, (, ) Gal(E, F ) E = F, but E is injective. Properties. For all (, ) Gal(E, F ), the closures Cl = E E and Cl = F F satisy 1) Cl and Cl are extensive. 2) and are antitone 3) Cl and Cl are monotone 4) =, and similarly = 5) Im = Im Cl = Fix Cl, called the set o closed elements o E 6) Cl Cl = Cl 7) ( strictly antitone) Inj Cl = Id E Im = E 8) x, x E, (x) (x ) (Im (x) (x )). 9) Denoting K = Im, K = Id K thus K is strictly antitone and K 1 = Im. Proos: 1) (x) (x) x ( (x)). 1) 2) : x, y E, x y ( (y)) (y) (x). 1) 2) 4) : Id E Cl Cl Cl =. 4) 5) : Cl = Im Cl Im. Then, Cl = Im Fix Cl Im Cl. Deductions 2) 3) and 4) 6) are obvious. 7) (Inj Cl = ) Cl = Id E (Im = E Cl = ) Cl = Id E strictly antitone. 8) (x) (x ) ( y F, x (y) x (y)). 9) K = Fix( ) Id K = Id K. Other proo : ( K, ) Gal(K, F ) with surjective. Finally, in Im K = Id K the roles o and are symmetrical. Remark. Properties 1) and 2) conversely imply that (, ) Gal(E, F ). Indeed, x E, y F, x (y) y ( (y)) (x). The properties o monotone Galois connections are deduced by reversing the order in F. instance i (u, v) Gal + (E, F ) then u and v are monotone, v u is extensive and u v Id F. Closure. A closure o an ordered set E is an E E such that, equivalently: 1) There exists a set F and an (u, v) Gal(E, F ) or Gal + (E, F ) such that v u = 2) is monotone, idempotent and extensive 3) x E, y Im, x y (x) y, i.e. (, Id K ) Gal + (E, K) where K = Im. Proo : 3) 1) 2); or 2) 3), x E, y K, x (x) y x y (x) (y) = y. Notes: The 2) 3) is a particular case o the above remark. Indeed, = Id K Id K (extensivity o the closure in K ). K E, K E, (, Id K ) Gal + (E, K) Im = K according to 7) with Id K injective. 11 For
SETS, RELATIONS, AND FUNCTIONS
September 27, 2009 and notations Common Universal Subset and Power Set Cardinality Operations A set is a collection or group of objects or elements or members (Cantor 1895). the collection of the four
More informationFoundations of mathematics
Foundations of mathematics 1. First foundations of mathematics 1.1. Introduction to the foundations of mathematics Mathematics, theories and foundations Sylvain Poirier http://settheory.net/ Mathematics
More informationINTRODUCTORY SET THEORY
M.Sc. program in mathematics INTRODUCTORY SET THEORY Katalin Károlyi Department of Applied Analysis, Eötvös Loránd University H1088 Budapest, Múzeum krt. 68. CONTENTS 1. SETS Set, equal sets, subset,
More informationMathematics for Computer Science/Software Engineering. Notes for the course MSM1F3 Dr. R. A. Wilson
Mathematics for Computer Science/Software Engineering Notes for the course MSM1F3 Dr. R. A. Wilson October 1996 Chapter 1 Logic Lecture no. 1. We introduce the concept of a proposition, which is a statement
More informationS(A) X α for all α Λ. Consequently, S(A) X, by the definition of intersection. Therefore, X is inductive.
MA 274: Exam 2 Study Guide (1) Know the precise definitions of the terms requested for your journal. (2) Review proofs by induction. (3) Be able to prove that something is or isn t an equivalence relation.
More informationIn mathematics you don t understand things. You just get used to them. (Attributed to John von Neumann)
Chapter 1 Sets and Functions We understand a set to be any collection M of certain distinct objects of our thought or intuition (called the elements of M) into a whole. (Georg Cantor, 1895) In mathematics
More informationSets and functions. {x R : x > 0}.
Sets and functions 1 Sets The language of sets and functions pervades mathematics, and most of the important operations in mathematics turn out to be functions or to be expressible in terms of functions.
More informationSets and Cardinality Notes for C. F. Miller
Sets and Cardinality Notes for 620111 C. F. Miller Semester 1, 2000 Abstract These lecture notes were compiled in the Department of Mathematics and Statistics in the University of Melbourne for the use
More informationCourse 221: Analysis Academic year , First Semester
Course 221: Analysis Academic year 200708, First Semester David R. Wilkins Copyright c David R. Wilkins 1989 2007 Contents 1 Basic Theorems of Real Analysis 1 1.1 The Least Upper Bound Principle................
More informationClassical Analysis I
Classical Analysis I 1 Sets, relations, functions A set is considered to be a collection of objects. The objects of a set A are called elements of A. If x is an element of a set A, we write x A, and if
More informationBasic Concepts of Set Theory, Functions and Relations
March 1, 2006 p. 1 Basic Concepts of Set Theory, Functions and Relations 1. Basic Concepts of Set Theory...1 1.1. Sets and elements...1 1.2. Specification of sets...2 1.3. Identity and cardinality...3
More informationChapter 3. Cartesian Products and Relations. 3.1 Cartesian Products
Chapter 3 Cartesian Products and Relations The material in this chapter is the first real encounter with abstraction. Relations are very general thing they are a special type of subset. After introducing
More informationChapter 1. Logic and Proof
Chapter 1. Logic and Proof 1.1 Remark: A little over 100 years ago, it was found that some mathematical proofs contained paradoxes, and these paradoxes could be used to prove statements that were known
More informationvertex, 369 disjoint pairwise, 395 disjoint sets, 236 disjunction, 33, 36 distributive laws
Index absolute value, 135 141 additive identity, 254 additive inverse, 254 aleph, 466 algebra of sets, 245, 278 antisymmetric relation, 387 arcsine function, 349 arithmetic sequence, 208 arrow diagram,
More informationA Note on Di erential Calculus in R n by James Hebda August 2010.
A Note on Di erential Calculus in n by James Hebda August 2010 I. Partial Derivatives o Functions Let : U! be a real valued unction deined in an open neighborhood U o the point a =(a 1,...,a n ) in the
More informationWeek 5: Binary Relations
1 Binary Relations Week 5: Binary Relations The concept of relation is common in daily life and seems intuitively clear. For instance, let X be the set of all living human females and Y the set of all
More informationMath 3000 Running Glossary
Math 3000 Running Glossary Last Updated on: July 15, 2014 The definition of items marked with a must be known precisely. Chapter 1: 1. A set: A collection of objects called elements. 2. The empty set (
More informationMAT2400 Analysis I. A brief introduction to proofs, sets, and functions
MAT2400 Analysis I A brief introduction to proofs, sets, and functions In Analysis I there is a lot of manipulations with sets and functions. It is probably also the first course where you have to take
More informationReview for Final Exam
Review for Final Exam Note: Warning, this is probably not exhaustive and probably does contain typos (which I d like to hear about), but represents a review of most of the material covered in Chapters
More informationThe Language of Mathematics
CHPTER 2 The Language of Mathematics 2.1. Set Theory 2.1.1. Sets. set is a collection of objects, called elements of the set. set can be represented by listing its elements between braces: = {1, 2, 3,
More informationAnalysis I: Calculus of One Real Variable
Analysis I: Calculus of One Real Variable Peter Philip Lecture Notes Created for the Class of Winter Semester 2015/2016 at LMU Munich April 14, 2016 Contents 1 Foundations: Mathematical Logic and Set Theory
More informationGROUPS ACTING ON A SET
GROUPS ACTING ON A SET MATH 435 SPRING 2012 NOTES FROM FEBRUARY 27TH, 2012 1. Left group actions Definition 1.1. Suppose that G is a group and S is a set. A left (group) action of G on S is a rule for
More informationLecture 16 : Relations and Functions DRAFT
CS/Math 240: Introduction to Discrete Mathematics 3/29/2011 Lecture 16 : Relations and Functions Instructor: Dieter van Melkebeek Scribe: Dalibor Zelený DRAFT In Lecture 3, we described a correspondence
More informationSets. A set is a collection of (mathematical) objects, with the collection treated as a single mathematical object.
Sets 1 Sets Informally: A set is a collection of (mathematical) objects, with the collection treated as a single mathematical object. Examples: real numbers, complex numbers, C integers, All students in
More information3. Equivalence Relations. Discussion
3. EQUIVALENCE RELATIONS 33 3. Equivalence Relations 3.1. Definition of an Equivalence Relations. Definition 3.1.1. A relation R on a set A is an equivalence relation if and only if R is reflexive, symmetric,
More informationDomain Theory: An Introduction
Domain Theory: An Introduction Robert Cartwright Rebecca Parsons Rice University This monograph is an unauthorized revision of Lectures On A Mathematical Theory of Computation by Dana Scott [3]. Scott
More informationProblem Set. Problem Set #2. Math 5322, Fall December 3, 2001 ANSWERS
Problem Set Problem Set #2 Math 5322, Fall 2001 December 3, 2001 ANSWERS i Problem 1. [Problem 18, page 32] Let A P(X) be an algebra, A σ the collection of countable unions of sets in A, and A σδ the collection
More informationIntroducing Functions
Functions 1 Introducing Functions A function f from a set A to a set B, written f : A B, is a relation f A B such that every element of A is related to one element of B; in logical notation 1. (a, b 1
More informationON GALOIS REALIZATIONS OF THE 2COVERABLE SYMMETRIC AND ALTERNATING GROUPS
ON GALOIS REALIZATIONS OF THE 2COVERABLE SYMMETRIC AND ALTERNATING GROUPS DANIEL RABAYEV AND JACK SONN Abstract. Let f(x) be a monic polynomial in Z[x] with no rational roots but with roots in Q p for
More informationBasic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011
Basic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011 A. Miller 1. Introduction. The definitions of metric space and topological space were developed in the early 1900 s, largely
More informationDiscrete Mathematics. Hans Cuypers. October 11, 2007
Hans Cuypers October 11, 2007 1 Contents 1. Relations 4 1.1. Binary relations................................ 4 1.2. Equivalence relations............................. 6 1.3. Relations and Directed Graphs.......................
More informationA set is a Many that allows itself to be thought of as a One. (Georg Cantor)
Chapter 4 Set Theory A set is a Many that allows itself to be thought of as a One. (Georg Cantor) In the previous chapters, we have often encountered sets, for example, prime numbers form a set, domains
More informationLecture 17 : Equivalence and Order Relations DRAFT
CS/Math 240: Introduction to Discrete Mathematics 3/31/2011 Lecture 17 : Equivalence and Order Relations Instructor: Dieter van Melkebeek Scribe: Dalibor Zelený DRAFT Last lecture we introduced the notion
More informationChapter 13: Basic ring theory
Chapter 3: Basic ring theory Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 42, Spring 24 M. Macauley (Clemson) Chapter 3: Basic ring
More informationPolynomials with nonnegative coefficients
Polynomials with nonnegative coeicients R. W. Barnard W. Dayawansa K. Pearce D.Weinberg Department o Mathematics, Texas Tech University, Lubbock, TX 79409 1 Introduction Can a conjugate pair o zeros be
More informationSome Definitions about Sets
Some Definitions about Sets Definition: Two sets are equal if they contain the same elements. I.e., sets A and B are equal if x[x A x B]. Notation: A = B. Recall: Sets are unordered and we do not distinguish
More informationChapter 1. Informal introdution to the axioms of ZF.
Chapter 1. Informal introdution to the axioms of ZF. 1.1. Extension. Our conception of sets comes from set of objects that we know well such as N, Q and R, and subsets we can form from these determined
More informationBasic Set Theory. Chapter Set Theory. can be written: A set is a Many that allows itself to be thought of as a One.
Chapter Basic Set Theory A set is a Many that allows itself to be thought of as a One.  Georg Cantor This chapter introduces set theory, mathematical induction, and formalizes the notion of mathematical
More informationThis chapter describes set theory, a mathematical theory that underlies all of modern mathematics.
Appendix A Set Theory This chapter describes set theory, a mathematical theory that underlies all of modern mathematics. A.1 Basic Definitions Definition A.1.1. A set is an unordered collection of elements.
More informationRevision of ring theory
CHAPTER 1 Revision of ring theory 1.1. Basic definitions and examples In this chapter we will revise and extend some of the results on rings that you have studied on previous courses. A ring is an algebraic
More informationPoint Set Topology. A. Topological Spaces and Continuous Maps
Point Set Topology A. Topological Spaces and Continuous Maps Definition 1.1 A topology on a set X is a collection T of subsets of X satisfying the following axioms: T 1.,X T. T2. {O α α I} T = α IO α T.
More informationSoftware Verification and Testing. Lecture Notes: Z I
Software Verification and Testing Lecture Notes: Z I Motivation so far: we have seen that properties of software systems can be specified using firstorder logic, set theory and the relational calculus
More informationSets, Relations and Functions
Sets, Relations and Functions Eric Pacuit Department of Philosophy University of Maryland, College Park pacuit.org epacuit@umd.edu ugust 26, 2014 These notes provide a very brief background in discrete
More informationTHE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS
THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS KEITH CONRAD 1. Introduction The Fundamental Theorem of Algebra says every nonconstant polynomial with complex coefficients can be factored into linear
More informationChapter 1 LOGIC AND PROOF
Chapter 1 LOGIC AND PROOF To be able to understand mathematics and mathematical arguments, it is necessary to have a solid understanding of logic and the way in which known facts can be combined to prove
More informationRelations Graphical View
Relations Slides by Christopher M. Bourke Instructor: Berthe Y. Choueiry Introduction Recall that a relation between elements of two sets is a subset of their Cartesian product (of ordered pairs). A binary
More informationModule MA1S11 (Calculus) Michaelmas Term 2016 Section 3: Functions
Module MA1S11 (Calculus) Michaelmas Term 2016 Section 3: Functions D. R. Wilkins Copyright c David R. Wilkins 2016 Contents 3 Functions 43 3.1 Functions between Sets...................... 43 3.2 Injective
More informationThe Consistency of the Continuum Hypothesis Annals of Mathematical Studies, No. 3 Princeton University Press Princeton, N.J., 1940.
TWO COMPUTATIONAL IDEAS Computations with Sets Union, intersection: computable Powerset: not computable Basic operations Absoluteness of Formulas A formula in the language of set theory is absolute if
More informationGeometric Transformations
Geometric Transformations Definitions Def: f is a mapping (function) of a set A into a set B if for every element a of A there exists a unique element b of B that is paired with a; this pairing is denoted
More informationChapter 10. Abstract algebra
Chapter 10. Abstract algebra C.O.S. Sorzano Biomedical Engineering December 17, 2013 10. Abstract algebra December 17, 2013 1 / 62 Outline 10 Abstract algebra Sets Relations and functions Partitions and
More informationFoundations of Mathematics I Set Theory (only a draft)
Foundations of Mathematics I Set Theory (only a draft) Ali Nesin Mathematics Department Istanbul Bilgi University Kuştepe Şişli Istanbul Turkey anesin@bilgi.edu.tr February 12, 2004 2 Contents I Naive
More informationNotes on Algebraic Structures. Peter J. Cameron
Notes on Algebraic Structures Peter J. Cameron ii Preface These are the notes of the secondyear course Algebraic Structures I at Queen Mary, University of London, as I taught it in the second semester
More informationIntroduction Russell s Paradox Basic Set Theory Operations on Sets. 6. Sets. Terence Sim
6. Sets Terence Sim 6.1. Introduction A set is a Many that allows itself to be thought of as a One. Georg Cantor Reading Section 6.1 6.3 of Epp. Section 3.1 3.4 of Campbell. Familiar concepts Sets can
More informationGraph Homomorphisms and Universal Algebra Course Notes
Graph Homomorphisms and Universal Algebra Course Notes Manuel Bodirsky, Institut für Algebra, TU Dresden, Manuel.Bodirsky@tudresden.de October 7, 2016 Contents 1 The Basics 2 1.1 Graphs and Digraphs................................
More informationit is easy to see that α = a
21. Polynomial rings Let us now turn out attention to determining the prime elements of a polynomial ring, where the coefficient ring is a field. We already know that such a polynomial ring is a UF. Therefore
More informationMA651 Topology. Lecture 6. Separation Axioms.
MA651 Topology. Lecture 6. Separation Axioms. This text is based on the following books: Fundamental concepts of topology by Peter O Neil Elements of Mathematics: General Topology by Nicolas Bourbaki Counterexamples
More informationLecture 4  Sets, Relations, Functions 1
Lecture 4 Sets, Relations, Functions Pat Place Carnegie Mellon University Models of Software Systems 17651 Fall 1999 Lecture 4  Sets, Relations, Functions 1 The Story So Far Formal Systems > Syntax»
More informationValid formulas, games and network protocols
Valid formulas, games and network protocols JeanLouis Krivine & Yves Legrandgérard Paris VII University, CNRS November 14, 2007 Introduction We describe a remarkable relation between a fundamental notion
More informationMathematics Review for MS Finance Students
Mathematics Review for MS Finance Students Anthony M. Marino Department of Finance and Business Economics Marshall School of Business Lecture 1: Introductory Material Sets The Real Number System Functions,
More informationDISCRETE MATHEMATICS W W L CHEN
DISCRETE MATHEMATICS W W L CHEN c W W L Chen, 1982, 2008. This chapter originates from material used by the author at Imperial College, University of London, between 1981 and 1990. It is available free
More informationPROBLEM SET 6: POLYNOMIALS
PROBLEM SET 6: POLYNOMIALS 1. introduction In this problem set we will consider polynomials with coefficients in K, where K is the real numbers R, the complex numbers C, the rational numbers Q or any other
More informationFoundations of Logic and Mathematics
Yves Nievergelt Foundations of Logic and Mathematics Applications to Computer Science and Cryptography Birkhäuser Boston Basel Berlin Contents Preface Outline xiii xv A Theory 1 0 Boolean Algebraic Logic
More informationThis asserts two sets are equal iff they have the same elements, that is, a set is determined by its elements.
3. Axioms of Set theory Before presenting the axioms of set theory, we first make a few basic comments about the relevant first order logic. We will give a somewhat more detailed discussion later, but
More information1 if 1 x 0 1 if 0 x 1
Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or
More informationChapter 1. SigmaAlgebras. 1.1 Definition
Chapter 1 SigmaAlgebras 1.1 Definition Consider a set X. A σ algebra F of subsets of X is a collection F of subsets of X satisfying the following conditions: (a) F (b) if B F then its complement B c is
More information! " # The Logic of Descriptions. Logics for Data and Knowledge Representation. Terminology. Overview. Three Basic Features. Some History on DLs
,!0((,.+#$),%$(&.& *,2($)%&2.'3&%!&, Logics for Data and Knowledge Representation Alessandro Agostini agostini@dit.unitn.it University of Trento Fausto Giunchiglia fausto@dit.unitn.it The Logic of Descriptions!$%&'()*$#)
More informationSECTION 6: FIBER BUNDLES
SECTION 6: FIBER BUNDLES In this section we will introduce the interesting class o ibrations given by iber bundles. Fiber bundles lay an imortant role in many geometric contexts. For examle, the Grassmaniann
More informationNominal Sets. c 2011 Andrew M. Pitts
Nominal Sets c 2011 Andrew M. Pitts Contents Preface page v 1 Permutations 1 1.1 The category of Gsets 1 1.2 Products and coproducts 3 1.3 Natural numbers 5 1.4 Functions 5 1.5 Power sets 6 1.6 Partial
More informationMathematics Course 111: Algebra I Part IV: Vector Spaces
Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 19967 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are
More informationCARDINALITY, COUNTABLE AND UNCOUNTABLE SETS PART ONE
CARDINALITY, COUNTABLE AND UNCOUNTABLE SETS PART ONE With the notion of bijection at hand, it is easy to formalize the idea that two finite sets have the same number of elements: we just need to verify
More informationPOWER SETS AND RELATIONS
POWER SETS AND RELATIONS L. MARIZZA A. BAILEY 1. The Power Set Now that we have defined sets as best we can, we can consider a sets of sets. If we were to assume nothing, except the existence of the empty
More informationAnnouncements. CompSci 230 Discrete Math for Computer Science Sets. Introduction to Sets. Sets
CompSci 230 Discrete Math for Computer Science Sets September 12, 2013 Prof. Rodger Slides modified from Rosen 1 nnouncements Read for next time Chap. 2.32.6 Homework 2 due Tuesday Recitation 3 on Friday
More informationThis chapter is all about cardinality of sets. At first this looks like a
CHAPTER Cardinality of Sets This chapter is all about cardinality of sets At first this looks like a very simple concept To find the cardinality of a set, just count its elements If A = { a, b, c, d },
More informationUnderstanding Basic Calculus
Understanding Basic Calculus S.K. Chung Dedicated to all the people who have helped me in my life. i Preface This book is a revised and expanded version of the lecture notes for Basic Calculus and other
More informationDiscrete Maths. Philippa Gardner. These lecture notes are based on previous notes by Iain Phillips.
Discrete Maths Philippa Gardner These lecture notes are based on previous notes by Iain Phillips. This short course introduces some basic concepts in discrete mathematics: sets, relations, and functions.
More informationMetric Spaces. Chapter 1
Chapter 1 Metric Spaces Many of the arguments you have seen in several variable calculus are almost identical to the corresponding arguments in one variable calculus, especially arguments concerning convergence
More informationABSTRACT ALGEBRA: A STUDY GUIDE FOR BEGINNERS
ABSTRACT ALGEBRA: A STUDY GUIDE FOR BEGINNERS John A. Beachy Northern Illinois University 2014 ii J.A.Beachy This is a supplement to Abstract Algebra, Third Edition by John A. Beachy and William D. Blair
More informationA Beginner s Guide to Modern Set Theory
A Beginner s Guide to Modern Set Theory Martin Dowd Product of Hyperon Software PO Box 4161 Costa Mesa, CA 92628 www.hyperonsoft.com Copyright c 2010 by Martin Dowd 1. Introduction..... 1 2. Formal logic......
More informationSMALL SKEW FIELDS CÉDRIC MILLIET
SMALL SKEW FIELDS CÉDRIC MILLIET Abstract A division ring of positive characteristic with countably many pure types is a field Wedderburn showed in 1905 that finite fields are commutative As for infinite
More information(Refer Slide Time: 1:41)
Discrete Mathematical Structures Dr. Kamala Krithivasan Department of Computer Science and Engineering Indian Institute of Technology, Madras Lecture # 10 Sets Today we shall learn about sets. You must
More informationINTRODUCTION TO TOPOLOGY
INTRODUCTION TO TOPOLOGY ALEX KÜRONYA In preparation January 24, 2010 Contents 1. Basic concepts 1 2. Constructing topologies 13 2.1. Subspace topology 13 2.2. Local properties 18 2.3. Product topology
More informationCHAPTER 5: MODULAR ARITHMETIC
CHAPTER 5: MODULAR ARITHMETIC LECTURE NOTES FOR MATH 378 (CSUSM, SPRING 2009). WAYNE AITKEN 1. Introduction In this chapter we will consider congruence modulo m, and explore the associated arithmetic called
More informationSet Theory. Pawel Garbacz
Set Theory Pawel Garbacz garbacz@kul.pl http://pracownik.kul.pl/garbacz/dydaktyka October 1, 2012 2 Contents 1 Baby set theory 5 1.1 Introduction.............................. 5 1.2 Further reading............................
More informationCHAPTER 1. Basic Ideas
CHPTER 1 asic Ideas In the end, all mathematics can be boiled down to logic and set theory. ecause of this, any careful presentation of fundamental mathematical ideas is inevitably couched in the language
More informationFinite Sets. Theorem 5.1. Two nonempty finite sets have the same cardinality if and only if they are equivalent.
MATH 337 Cardinality Dr. Neal, WKU We now shall prove that the rational numbers are a countable set while R is uncountable. This result shows that there are two different magnitudes of infinity. But we
More informationCS 580: Software Specifications
CS 580: Software Specifications Lecture 2: Sets and Relations Slides originally Copyright 2001, Matt Dwyer, John Hatcliff, and Rod Howell. The syllabus and all lectures for this course are copyrighted
More informationG = G 0 > G 1 > > G k = {e}
Proposition 49. 1. A group G is nilpotent if and only if G appears as an element of its upper central series. 2. If G is nilpotent, then the upper central series and the lower central series have the same
More informationNotes. Sets. Notes. Introduction II. Notes. Definition. Definition. Slides by Christopher M. Bourke Instructor: Berthe Y. Choueiry.
Sets Slides by Christopher M. Bourke Instructor: Berthe Y. Choueiry Spring 2006 Computer Science & Engineering 235 Introduction to Discrete Mathematics Sections 1.6 1.7 of Rosen cse235@cse.unl.edu Introduction
More informationProblem Set 1 Solutions Math 109
Problem Set 1 Solutions Math 109 Exercise 1.6 Show that a regular tetrahedron has a total of twentyfour symmetries if reflections and products of reflections are allowed. Identify a symmetry which is
More informationNo: 10 04. Bilkent University. Monotonic Extension. Farhad Husseinov. Discussion Papers. Department of Economics
No: 10 04 Bilkent University Monotonic Extension Farhad Husseinov Discussion Papers Department of Economics The Discussion Papers of the Department of Economics are intended to make the initial results
More informationMethoδos Primers, Vol. 1
Methoδos Primers, Vol. 1 The aim of the Methoδos Primers series is to make available concise introductions to topics in Methodology, Evaluation, Psychometrics, Statistics, Data Analysis at an affordable
More information1.1 Logical Form and Logical Equivalence 1
Contents Chapter I The Logic of Compound Statements 1.1 Logical Form and Logical Equivalence 1 Identifying logical form; Statements; Logical connectives: not, and, and or; Translation to and from symbolic
More informationDiscrete Mathematics Set Operations
Discrete Mathematics 13. Set Operations Introduction to Set Theory A setis a new type of structure, representing an unordered collection (group, plurality) of zero or more distinct (different) objects.
More informationIntroduction to Algebraic Geometry. Bézout s Theorem and Inflection Points
Introduction to Algebraic Geometry Bézout s Theorem and Inflection Points 1. The resultant. Let K be a field. Then the polynomial ring K[x] is a unique factorisation domain (UFD). Another example of a
More information(LMCS, p. 317) V.1. First Order Logic. This is the most powerful, most expressive logic that we will examine.
(LMCS, p. 317) V.1 First Order Logic This is the most powerful, most expressive logic that we will examine. Our version of firstorder logic will use the following symbols: variables connectives (,,,,
More informationMATH 433 Applied Algebra Lecture 13: Examples of groups.
MATH 433 Applied Algebra Lecture 13: Examples of groups. Abstract groups Definition. A group is a set G, together with a binary operation, that satisfies the following axioms: (G1: closure) for all elements
More informationPredicate Logic Review
Predicate Logic Review UC Berkeley, Philosophy 142, Spring 2016 John MacFarlane 1 Grammar A term is an individual constant or a variable. An individual constant is a lowercase letter from the beginning
More informationREAL ANALYSIS I HOMEWORK 2
REAL ANALYSIS I HOMEWORK 2 CİHAN BAHRAN The questions are from Stein and Shakarchi s text, Chapter 1. 1. Prove that the Cantor set C constructed in the text is totally disconnected and perfect. In other
More informationYou know from calculus that functions play a fundamental role in mathematics.
CHPTER 12 Functions You know from calculus that functions play a fundamental role in mathematics. You likely view a function as a kind of formula that describes a relationship between two (or more) quantities.
More informationDiscrete Mathematics Lecture 5. Harper Langston New York University
Discrete Mathematics Lecture 5 Harper Langston New York University Empty Set S = {x R, x 2 = 1} X = {1, 3}, Y = {2, 4}, C = X Y (X and Y are disjoint) Empty set has no elements Empty set is a subset of
More information