Review for Final Exam


 Kathleen Ethel Owens
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1 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 111 inclusive that we covered in class or on the assignments. In each section, review by reading over the material, and looking over the corresponding assignment or odd numbered problems in the text. This is an expanded version of the Midterm II review. 1 Chapter 1: Sets Definition 1 A set is a collection of objects. Its members are called elements. We usually denote sets with capital letters and element with lowercase letters. a A iff a is an element of the set A. a A iff a is not an element of A. Note: If there are a finite number of elements in a set A, then the order of elements in the list is irrelevant. Also an element can only appear once in any set. Definition 2 The empty set or null set is denoted by. It is the set with no elements. 1.1 Describing Sets We can describe sets with a finite number of elements by just listing them. i.e A = {a 1,..., a n }. We can also sometimes just describe sets by listing elements if the pattern they follow is obvious. eg. N = {1, 2, 3,...}. In other cases, we might want a new approach. A common one is to write S = {x : p(x)}. Here S is the set of all x satisfying property p(x). For commonly used sets, we try to use the same names. Here are some examples: N = {1, 2, 3,...} is the set of natural numbers. Z = {..., 3, 2, 1, 0, 1, 2, 3,...} is the set of integers. Q = { a b : a, b Z, b 0} is the set of rational numbers. R is the set of real numbers and I = {x R : x Q} is the set of irrational numbers. 1
2 C = {a + bi : a, b R} is the set of complex numbers where i = 1. a < b both real numbers define a bunch of intervals on the real line. (a, b) = {x R : a < x < b} is the open interval. [a, b] = {x R : a x b} is the closed interval. (a, b] = {x R : a < x b} and [a, b) = {x R : a x < b} are halfclosed intervals. Definition 3 A set S is called finite if it has a finite number of elements. If S has n elements, we write S = n and say S has cardinality S. If S is not finite, it is called infinite. We won t define the cardinality of an infinite set until Chapter Subsets and Set Operations Definition 4 Let A, B be 2 sets. A is a subset of B, written A B iff x A implies x B. Note that A and A A for all sets A. Two sets A and B are equal, A = B, if they have exactly the same elements. Note that A = B if and only if A B and B A. A is a proper subset of B, written as A B if A B but A B. A universal set is just a set which contains a bunch of other sets under discussion as subsets. A Venn diagram is a diagram of sets. Usually the universal set is depicted as a rectangle and the sets inside are depicted as circles. It is a useful picture of set relations. P(A) = {X : X A} is the power set of A consisting of all subsets of a set A. Warning: A set can be both an element of another set and a subset of another set. It is important to note how many brackets are involved. Eg. A = {a, {a}}. Here {a} A and {a} A. Definition 5 Let A, B be sets, subsets of some universal set U. 2
3 The intersection of A and B is The union of A and B is The complement of A is is A B = {x : x A and x B} A B = {x : x A or x B} A = {x U : x A} The relative complement or set difference of A and B is The Cartesian product of A B is A B = {x : x A and x B} A B = {(x, y) : x A and y B} Venn diagrams are useful for picturing the above concepts. Results on Cardinalities Let A, B be finite sets. Then If A B =, A B = A + B. In general, A = A B + A B and A B = A + B A. P(A) = 2 A. A B = A B. If 0 k n = A, then for A(k) = {X P(A) : X = k}, ( ) n n! A(k) = = k (n k)!k! 3
4 1.3 Indexed Collections of Sets and Partitions of Sets Definition 6 Let I be an index set (finite or infinite) and let {S α : α I} be a collection of sets. The union of {S α : α I} is the set α I S α = {x : x S α for some α I} The intersection of {S α : α I} is the set α I S α = {x : x S α for all α I} Definition 7 Two sets A, B are called disjoint if A B =. A collection of sets S is called pairwise disjoint if each pair of distinct sets in the collection are disjoint. Definition 8 A partition of a nonempty set A can be defined in a number of equivalent ways. Here are A partition of A is a collection S of pairwise disjoint nonempty subsets of A with the property that each element of A belong to some set in S. 2. A partition of A is a collection S of nonempty subsets of A such that every element of A belongs to exactly one set in S. 3. A partition of A is a collection S of subsets of A satisfying: X if X S. X, Y S implies X = Y or X Y =. X S X = A. Note that the number of parts or the size of the partition is the number of subsets in the collection S above. 4
5 2 Chapter 2: Logic Definition 9 A statement is a declarative sentence which is either true or false, not both. Each statement has a truth value: T if true and F if false. We use capital letters P, Q R to denote statements or P i, Q i, R i, i = 1,..., n. An open statement over a domain S is a statement P (x) for each x S. This could be written as P (x 1,..., x n ) if the domain S = S 1 S 2 S n. A truth table for statements P 1,..., P n is a table with 2 n rows that records all 2 n possible combinations of truth values of the statements P 1,..., P n. Usually the first n statements are independent statements and the subsequent statements R 1,..., R m all depend on the truth values of P 1,..., P n. 2.1 Combining Statements into New Statements Definition 10 Let P, Q be statements. The following are statements that are dependent on P and/or Q. The book calls these compound statements. The negation of P is not P and is written as P. Its truth values are the opposite of those of P. The truth table is in Fig 2.2 of the text. The disjunction of P and Q is the statement P or Q and is written P Q. It is true if and only if at least one of P or Q is true. See Figure 2.3. The conjunction of P and Q is the statement P and Q and is written P Q. It is true if and only if both P and Q are true. See Figure 2.4. The implication is the statement P Q. Its most common description is If P, then Q. But there are other equivalent expressions in English which you should know: If P then Q; Q if P ; P implies Q; P only if Q; 5
6 P is sufficient for Q; Q is necessary for P. P Q has the most surprising truth table. In fact P Q is only false if P is true and Q is false. This is extremely important. See Figure 2.5. The implication Q P is called the converse of P Q. The biconditional P Q is P is equivalent to Q or P if and only if Q or P is a necessary and sufficient condition for Q. The operations,,,, are logical connectives. Note: One could do any of the above operations for each statement in an open sentence over a domain S. eg. P (x) Q(x) for x S etc. Then one would have to decide on the truth value of the compound statement for each x S. Definition 11 Let P 1,..., P n be independent statements. A tautology is a compound statement R depending on P 1,..., P n such that for all 2 n choices of the truth values of P 1,..., P n, the truth value of R is true. A contradiction is a compound statement R depending on P 1,..., P n such that for all 2 n choices of the truth values of P 1,..., P n, the truth value of R is false. Two compound statements R and S depending on P 1,..., P n are logically equivalent, written R S if and only if for all 2 n choices of the truth values of P 1,..., P n, the truth values of R and S are the same. You can then prove that R S by simply checking their values in a truth table are equal. 2.2 Results on Logical Equivalence Instead of using a truth table to prove compound statements are logically equivalent, you can combine the following results that were already verified by truth table. Theorem 1 For statements P, Q and R, 6
7 1. Commutative Laws (a) P Q Q P (b) P Q Q P. 2. Associative Laws (a) P (Q R) (P Q) R (b) P (Q R) (P Q) R. 3. Distributive Laws (a) P (Q R) (P Q) (P R) (b) P (Q R) (P Q) (P R). 4. DeMorgan s Laws (a) (P Q) ( P ) ( Q) (b) (P Q) ( P ) ( Q). The following is very important although it really just summarises the truth values of the implication. Theorem 2 For statements P, Q, P Q ( P ) Q. Its negation is a consequence: (P Q) P ( Q). P Q (P Q) (Q P ). P Q ( Q) ( P ) (this is the contrapositive of the first implication). Any other logical equivalences can be derived from those above rules. 2.3 Quantified Statements We can convert an open sentence over a domain, P (x), x S into a single statement in 2 ways. This process is called quantification. Definition 12 Let P (x), x S be a statement for each x S. Then 1. The universal quantifier gives the statement x S, P (x) which translates as: For all x S, P (x) is true. 7
8 2. The existential quantifier gives the statement x S, P (x) which translates as: There exists x S such that P (x) is true. It is important to be able to translate back and forth from English to logic. The only other result in this section is: Theorem 3 Let P (x) be a statement for each x S. 1. ( x S, P (x)) ( x S, (P (x))) 2. ( x S, P (x)) ( x S, (P (x))) Loosely speaking, this means that these quantifiers are opposites. Note that this continues to work if there is more than one variable involved. That is Theorem 4 Let P (x 1, x 2,..., x n ) be a statement for each x i S i. 1. ( x i S i, P (x 1,..., x n )) ( x i S i, (P (x 1,..., x n ))) 2. ( x i S i, P (x 1,..., x n )) ( x i S i, (P (x 1,..., x n ))) You should be able to translate quantified statements and their negations between English and logic and back. Definition 13 x S, P (x) Q(x) means that open sentence P (x) is characterised by open sentence Q(x) over the domain S. 3 Chapter 3 In Chapter 3, we learn proof techniques to prove the statement x S, P (x) Q(x) where P (x), Q(x) are statements for each x S. 1. Trivial Proof : In this method, we simply prove that Q(x) is true for all x S. Then the implication P (x) Q(x) is true for all x S, no matter the truth value of P (x). 8
9 2. Vacuous Proof : In this method, we simply prove that P (x) is false for all x S. Then the implication P (x) Q(x) is true for all x S, no matter the truth value of Q(x). 3. Direct Proof : In this most common method of proof, we assume for each x S, P (x) is true and deduce that Q(x) is true. 4. Proof by Contrapositive: In this useful method of proof, we prove instead the contrapositive: i.e. x S, (Q(x)) (P (x)). That is, for each x S, we assume Q(x) is false and deduce that then P (x) is false. 5. Proof by Cases: Here, instead of proving x S, P (x) Q(x), we first write S = n i=1s i and for each i = 1,..., n, we prove x S i, P (x) Q(x) by some method of proof mentioned earlier (or even proof by contradiction, which we didn t do until Ch 5). Note that all the proofs in this chapter were about even and odd integer and parity. Definitions are given in the section on Chapter 4 which also talks about more situations to which one might apply these proof techniques. 4 Chapter 4 In Chapter 4, we use the 3 main proof techniques of Chapter 3: Direct Proof, Proof by Contrapositive, Proof by Cases to prove results of 3 main types: 1. Divisibility of Integers and Congruence Modulo a positive integer 2. Inequalities and Absolute Value 3. Set Operations including Cartesian Products 4.1 Divisibility of Integers and Congruence Definition 14 For a, b Z, a 0, a divides b iff b = ka for some k Z. We write a b and also say that b is a multiple of a or a is a divisor of b. We write a b if a doesn t divide b. Note that for a Z, 2 a iff a is even. Its contrapositive says 2 a iff a is odd. 9
10 Results: 1. (Transitivity) Let a, b, c Z, a 0, b 0. If a b and b c then a c. 2. Let a, b, c, d Z and a 0, c 0. Then ac bd. 3. Let a, b, c, x, y Z and a 0. If a b and a c then a (bx + cy) (i.e. a divides b and c implies that a divides any linear combo of b and c.) Theorem 5 For integers x, y Z, 2 xy if and only if 2 x or 2 y. Corollary: For x Z, 2 x 2 if and only if 2 x. Definition 15 A prime number is a positive integer p > 1 whose only positive divisors are 1 and p. Note: Although we will not prove this until Ch 11, it is in fact true that if p is a prime, for integers x, y Z, p xy if and only if p x or p y. This implies that for x Z, p x 2 if and only if p x. We did prove this for p = 2, 3, 5 by case work. Definition 16 For a, b Z and n N, n 2, we say a is congruent to b modulo n, written a b mod n iff n (a b). Note that for n = 2, a b mod 2 means that a and b have the same parity (i.e. are both even or both odd). If a b mod 2 we say they have opposite parity. Basic Results: Let a, b, c, d, n Z, n (Reflexive) a a mod n. 2. (Symmetric) If a b mod n then b a mod n. 3. (Transitive) If a b mod n and b c mod n then a c mod n. 4. (Additive) If a b mod n and c d mod n then a + c b + d mod n. 5. (Multiplicative) If a b mod n and c d mod n then ac bd mod n. 10
11 4.2 Inequalities and Absolute Value Basic Results: Let x, y, z R. 1. Exactly one of the following is true: x < y,x = y or x > y. 2. (Antisymmetric) x y and y x implies that x = y. 3. (Transitive) x y and y z implies that x z. 4. (Additive) x y implies that x + z y + z. 5. If x y and z > 0 then xz yz. 6. If x y and z < 0 then xz yz. 7. If 0 < x y then 0 < 1 y 1 x. Definition 17 The absolute value of x R is written as x. x = x if x 0 and x = x if x < 0. Results: Let x, y R. 1. x 0 and x 2 = x. 2. xy = x y. 3. (Triangle inequality) x + y x + y. Result: x, y R, xy = 0 implies x = 0 or y = Set Operations including Cartesian Products See Chapter 1 for definitions of Set Operations. Result: A, B sets. A B = iff A = or B =. Result: A, B sets in some universal set U. Then A B and B A implies A = B. There are many set identities. But the above results and the following theorem (which is a direct analogue of the logic laws), are sufficient to prove any other set identity. Theorem 6 For sets A, B, C 1. (Commutativity) 11
12 (a) A B = B A (b) A B = B A. 2. (Associativity) (a) A (B C) = (A B) C (b) A (B C) = (A B) C 3. (Distributivity) (a) A (B C) = (A B) (A C) (b) A (B C) = (A B) (A C) 4. (DeMorgan s Laws) (a) A B = A B (b) A B = A B 5 Chapter 5 In Chapter 5, you learned the following proof techniques: 1. Counterexamples 2. Proof by Contradiction 3. Existence Proofs 4. Nonexistence Proofs 5.1 Counterexamples Let P (x) be a statement for every x S. In this section, you need to disprove x S, P (x) by proving its negation x S, P (x). That is you need to find a counterexample in S, i.e. an element x of S for which P (x) is false. 5.2 Proof by Contradiction Here we prove a statement R by assuming that R is false, and showing that R implies a contradiction C. 12
13 Then, in symbols, we showed R C. But since C is a contradiction, C is false and R C is true then (looking at the truth table) we see that R must be false so that R is true. R above could be a quantified statement such as x S, P (x) or x S, P (x) Q(x), but in any case, we assume its negation R and find a contradiction to the hypotheses or known facts. The applications in this section use results from Chapter 4 (divisibility, congruence, real numbers, inequalities, set theory) or the following new application to rational versus irrational numbers: Recall that Q = { x : x, y Z, y 0} is the set of rationals and I = R Q y are the irrationals. Note that any x Q can be expressed as p where p and y q q are integers with no common positive integer divisor. Note that Q is closed under all arithmetic operations: That is if x, y Q, then x + y Q, x y Q, xy Q and x/y Q if also y 0. The same goes for the real numbers R. That is if x, y R, then x + y R, x y R, xy R and x/y R if also y 0. Contradiction gives a method to prove that certain numbers are irrationals. To prove a real number is irrational by contradiction, assume it is rational and find a contradiction. Theorem 7 2 I This was proved by contradiction, by assuming 2 = x/y where x/y is in lowest terms. The contradiction is obtained by using the fact that 2 x 2 implies 2 x and by contradicting the fact that x/y was in lowest terms. That is, by finding that 2 is a common divisor of x and y. Using the fact that for a prime p and x Z, p x 2 implies that p x, one could imitate this proof to show that p I for any prime p. In the assignment, you imitated this proof to show that 2n I if n Z is odd. Combining rationals and irrationals with arithmetic turns out badly for the rationals. One can use contradiction to prove: Result: x Q, a I, then x + a I, xa I, x/a I. 5.3 Existence Proofs Let S be a set. Let P (x) be a statement for each x S. Here we are proving x S, P (x). 13
14 Of course, one could always produce an example of an element x of S such that P (x) is true. This section points out that sometimes one can prove an existence statement as above without giving an explicit example. One example is Result: There exist a, b I such that a b Q. The proof takes a = b = 2 but does not decide whether or not a b is irrational, it just proves that if not, there exists c, d I with c d rational. Another example is the use of the Intermediate Value theorem in Calculus to find roots of continuous functions: Theorem 8 (IVT) If a < b are real numbers, and f : [a, b] R is a continuous function, then for any k between f(a) and f(b) there exists c (a, b) such that f(c) = k. This may be applied to find real roots to continuous functions (such as polynomial functions). For a continuous function f : R R, just find a, b R with f(a) < 0 < f(b) and then there must be a c between a and b with f(c) = 0. One could also use contradiction to show the uniqueness of such a root. See eg, 5.23 in the text. Existence proofs work sometimes well to prove statements about rationality or irrationality. 5.4 Disproving Existence Statements To disprove x S, P (x) just prove its negation: x S, P (x). No new techniques occur in this section. 6 Chapter 6: Induction Definition 18 A subset S of R is wellordered if every subset of S has a least element. WellOrdering Principle N is wellordered. Other examples of wellordered sets include: finite subsets of R, subsets of wellordered sets, {x Z : x a} for some a Z. Nonexamples of wellordered sets include: Z, Q, R, [a, b]. This is the basis of the Principle of Mathematical Induction: Theorem 9 For a fixed m Z, let S = {i Z : i m}. For each n S, let P (n) be a statement. If 14
15 P (m) is true, and k S, P (k) P (k + 1) is true then n S, P (n) is true. This principle is useful for proving statements of the following form: For fixed m Z, let S = {i Z : i m} 1. Summation formulas, eg. 6.10, 6.38 from HW. 2. Inequalities that hold for n S., eg from HW, 6.14 from class. 3. Divisibility statements for integers that hold for all n S, eg from HW. 4. Extending results that are known for n = 2 to a result for all positive integers n 2.: eg from A6. There is a stronger form of the principle of mathematical induction which is useful for some statements: Theorem 10 (Strong Induction) For a fixed m Z, let S = {i Z : i m}. For each n S, let P (n) be a statement. If P (m) is true, and k S, P (m) P (m + 1) P (k) P (k + 1) is true then n S, P (n) is true. The main difference is that in the inductive step you get to assume that all the previous statements P (r), m r k are true, in order to prove that P (k + 1) is true. Strong induction is useful for proving statements of the following form: 1. Proving general formulas for recursion relations, eg from A6. 2. Showing that for 2 positive relatively prime integers a, b (i.e. two positive integers with only positive common divisor 1) that there exists m N such that for all n m, there exist x, y N such that ax + by = n. (eg. 6.44, or making change question from class.) 15
16 3. This allowed us to prove: For every 1 n N, n can be written as a product of prime numbers. 4. Geometric problems: such as the number of regions bounded by n lines in general position (done in class), the sum of the interior angles of an ngon n 3 (done in class and 6.51), and Additional Problem 2 on A6. 5. Additional Problem 1 on A6. One other point from Chapter 6: As above, summation formulas over N can be proved by induction. One can use rules about summations to combine known formulas to get new ones. Summation Results: 1. Let a i, b i R for all i = 1,..., n for some n N. Let c R. Then n i=1 (a i + cb i ) = n i=1 a i + c( n i=1 b i). 2. n i=1 c = nc for c R. 3. n i=1 i = n(n+1) 2 for all n N. 4. n i=1 i2 = n(n+1)(2n+1) 6 for all n N. 7 Chapter is interesting to read as it talks about famous conjectures in mathematics, such as Fermat s Last Theorem, and the 4 colour problem. In 7.2, the topic is mixing the quantifiers and. Let P (x, y) be a statement for every x S and y T. We look at and x S, y T, P (x, y) x S, y T, P (x, y) The main points are: Never replace a by a or vice versa. The order of the quantifiers often changes the statement. usually, we have: That is, x S, y T, P (x, y) y T, x S, P (x, y) 16
17 The negations of the statements above are: and ( x S, y T, P (x, y)) x S, y T, P (x, y) ( x S, y T, P (x, y)) x S, y T, P (x, y) Be able to translate quantified statements symbolically, as in 7.8, 7.12 and to find their negations. The remaining 2 sections just cover Prove or Disprove a given statement. In each case, you must decide whether the statement is true or false, and prove either the statement if true, or its negation if false. No new techniques or facts are presented here, but it is good practice to look through some questions in this section to see if you understand the previous sections. 8 Equivalence Relations Definition 19 Let A and B be sets. A relation R from A to B is a subset of A B. That is R A B. The domain of R is the set The range of R is the set dom(r) = {a A : (a, b) R} ran(r) = {b B : (a, b) R} Note dom(r) A and ran(r) B. If (a, b) R, we write a R b and if (a, b) R, we write a R b. A relation on a set S is a relation from S to S or a subset of S S. Properties of Relations A relation R on a set S might have some of the following properties. R is reflexive if for all a S, a R a. R is symmetric if for a, b S, a R b b R a. R is transitive if for a, b, c S, (a R b) (b R c) a R c. 17
18 R is antisymmetric if for a, b S, (a R b) (b R a) a = b. Most of these properties are easy to check. The one that can be a bit tricky in finite cases is transitivity. Note that what you actually have to prove is that for all pairs (a, b) R, (b, c) R with a b and b c, we have (a, c) R. Definition 20 An equivalence relation on S is a relation R which is reflexive,symmetric and transitive. A partial order on S is a relation R which is reflexive,antisymmetric and transitive. Examples of Equivalence Relations S any set; for a, b S, a R b a = b. S = Z and let n Z, n 2. For a, b Z, a R b a b mod n. S = R R, (a, b), (c, d) S, (a, b) R (c, d) a 2 + b 2 = c 2 + d 2 Definition 21 For an equivalence relation R on S, the equivalence class of a S is [a] = {x S : x Ra} Theorems on Equivalence Relations and Classes Let R be a equivalence relation on a set S. 1. Let a, b S. Then a R b iff [a] = [b]. 2. Equivalence relations on S are in 11 correspondence with partitions of S. More precisely, if R is an equivalence relation on S. Then the set of distinct equivalence classes {[a i ] : i I} forms a partition of S. That is, [a i ] [a j ] = if i j. and S = i I [a i ]. Conversely if S has a partition {S α : α I} then R = α I S α S α is an equivalence relation with distinct equivalence classes {S α : α I}. [Here a R b iff a S α and b S α for some α I.] 3. Note that for a, b S, either [a] [b] = (if (a, b) R) or [a] = [b] (if (a, b) R). 18
19 4. To find distinct equivalence classes {[a i ] : i I} of an equivalence relation R on S, you need to show that if x S, then [x] = [a i ] for some i I (i.e. that x R a i ) and that [a i ] [a j ] = if i j (i.e. that (a i, a j ) R). Results for the Congruence Modulo n Equivalence Relation Let a, b Z and let n N, n 2. Set a R b iff a b mod n 1. The distinct equivalence classes are {[k] : 0 k n 1} Note that [k] = {nq + k : n Z} under this equivalence relation. These sets then partition Z. 2. Let Z n = {[x] : x Z} = {[k] : 0 k n 1}. We may define an addition and multiplication on Z n by setting x, y Z : [x] + [y] := [x + y], [x][y] := [xy] Note, that it is necessary to check that these operations on Z n are welldefined. That is, for x, x, y, y Z with [x] = [x ] and [y] = [y ], then check [x] + [y] = [x ] + [y ] and [x][y] = [x ][y ]. This follows from the additive and multiplicative properties of congruence modulo n which shows that [x + y] = [x + y ] and [xy] = [x y ]. Examples of Partial Orders S R. For x, y S, x R y iff x y. S N, For x, y S, x R y iff x y S = P(X) for some set X. For A, B S, A R B iff A B. For a partial order R on a finite set S, one can draw a picture of the partial order called a Hasse diagram. Let S be the vertices of the diagram. Connect a with a downward pointing arrow to b iff a R b and there does not exist a c S with a R c and c R b. 9 Chapter 9: Functions Definition 22 Let A, B be nonempty sets. Then a function f from A to B is a relation f with f({a}) = 1 for all a A. (See definition under relations: f({a}) = {b B : (a, b) f}.) That is, for each a A, there is exactly one b B with (a, b) f. This b is called f(a) or the image of a under f. We will later write a function as a map f : A B. 19
20 Definition 23 Looking again at the definitions we made for relations: For a function f : A B and subsets A 0 A and B 0 B, f(a 0 ) = {f(a 0 ) B : a 0 A 0 } is the image of A 0 under f. In particular, the image of f is f(a). It is a subset of B and f 1 (B 0 ) = {a A : f(a) B 0 } is the preimage of B 0 under f. In particular, we will be interested in the fibre of b B under f given by f 1 ({b}) = {a A : f(a) = b}. Definition 24 (Properties of functions) Let f : A B be a function. f is 11 or injective iff for all x A, y A such that x y then f(x) f(y). That is distinct elements of A have distinct images under f in B. The contrapositive of this definition is very useful: f is 11 or injective iff for all x, y A, f(x) = f(y) implies that x = y. Another equivalent formulation is: f is 11 or injective iff for all b B, f 1 ({b}) 1. f is onto or surjective iff f(a) = B iff for all b B, there exists a A such that f(a) = b iff for all b B, f 1 ({b}) 1. Note that the 2nd formulation is what you usually use to check f is surjective. It is actually equivalent to B f(a) as f(a) B is automatic for any function. f is bijective iff f is surjective and injective. Note then that for all b B, f 1 ({b}) = 1. Lemma 1 A function f : R R is injective if it is either strictly increasing (i.e. x < y implies that f(x) < f(y)) or strictly decreasing (i.e. x < y implies that f(x) > f(y). Theorem 11 Let A and B be finite nonempty sets. Let f : A B be a function. Then (a) f(a) A and f(a) B. 20
21 (b) f is injective iff f(a) = A. So f is injective implies that A B (c) f is surjective iff f(a) = B. So f is surjective implies that A B. (d) (Pigeonhole principle) Now suppose A = B. Then f : A B is bijective iff f is injective iff f is surjective. Definition 25 A relation R from A to B gives an inverse relation R 1 from B to A given by R 1 = {(b, a) : (a, b) R} B A Definition 26 A function f : A B is called invertible iff f 1 is a function from B to A. Then f 1 : B A satisfies f 1 (b) = a iff f(a) = b. Theorem 12 Let A and B be nonempty sets. Then f : A B is bijective if and only if f is invertible. If so, f 1 : B A is the unique function such that f 1 (b) = a if and only if f(a) = b. Definition 27 The composite of two functions f : A B and g : B C is a function g f : A C such that (g f)(a) = g(f(a)) for a A. i A : A A is the identity function on A with i A (a) = a. Theorem 13 (Composites and inverses) Let f : A B, g : B C and h : C D be functions. 1. f i A = f and i B f = f. 2. (Associativity of composition) f (g h) = (f g) h. 3. If f : A B is a bijection then its inverse f 1 : B A satisfies f 1 f = i A and f f 1 = i B. If g : B A is another function such that g f = i A and f g = i B then g = f 1. So f 1 is unique. 4. If f and g are injective, then so is g f : A C. 5. If f and g are surjective, then so is g f : A C. 6. If f and g are bijective, then so is g f : A C. The inverse of g f is then (g f) 1 = f 1 g If f is bijective, then so is f 1. The inverse of f 1 is (f 1 ) 1 = f. 21
22 Definition 28 The set of permutations on the set I n = {1,..., n} is S n = {f : I n I n : f is bijective} Note that S n = n. Each element of n can be written as ( ) 1 2 n 1 n f = f(1) f(2) f(n 1) f(n) This notation makes it easy to compute for α, β S n α β S n and β 1 S n since α β(i) = α(β(i)) and β 1 (j) = i iff β(i) = j. 10 Chapter 10: Cardinality of Sets Theorem 14 Let S be a nonempty collection of nonempty sets. A relation R is defined on S by A R B iff there exists a bijective function from A to B. Then R is an equivalence relation on S. Definition 29 Two sets A and B have the same cardinality, written A = B, iff either A = B = or there exists a bijective function from A to B. We also say that A and B are numerically equivalent sets. Otherwise, A B. Note that for the equivalence relation given in the previous theorem, the equivalence class of A consists of all elements of with the same cardinality as B Denumerable Sets Definition 30 A set A is called denumerable iff A = N, that is iff there exists a bijective function f : N A. A set A is called countable iff A is either denumerable or finite. A set A is called uncountable iff A is not countable. Theorem 15 (Results about Denumerable Sets) 1. Z is denumerable 2. Q + and Q are denumerable. 3. Every infinite subset of a denumerable set is denumerable. 4. If A and B are denumerable sets, then A B is denumerable. 22
23 10.2 Uncountable Sets Theorem 16 The open interval (0, 1) is uncountable. Proposition 1 If A B are sets and A is uncountable, then B is uncountable. A consequence of this theorem is Corollary 1 R is uncountable But Theorem 17 (0, 1) and R are numerically equivalent. In fact it is true from your assignment 9 and class results that any nontrivial interval (of finite or infinite length, open or closed) is numerically equivalent to (0, 1) and so also to R. Recall that for a set A, 2 A = {f : A {0, 1} : f is a function} Note that the below theorem is clear for finite sets A as 2 A = 2 A = P(A) in the case A <. Theorem 18 For a set A, 2 A and P(A) are numerically equivalent. Definition 31 For sets A, B, we say that A B iff there exists a 11 function g : A B. Note that this generalises the result for finite sets A and B. Theorem 19 Let A and B be nonempty sets. (a) If f : A B is an injective function, then there exists a function g : B A (called a left inverse) with g f = i A. This function g is surjective. (b) If h : B A is surjective, then there exists a function k : A B (called a right inverse) with h k = i A. Then k is injective. (c) There exists an injective function f : A B iff there exists a surjective function g : B A. This means that we can also find the analogous definition: 23
24 Corollary 2 For nonempty sets A and B, A B iff there exists a surjective function g : A B The book emphasises A < B which is of course equivalent to A B and A B or that for sets A and B there exists an injective function from A to B but there does not exist a bijection between A and B. Equivalently, from the corollary, there exists a surjective function g : B A but there does not exist a bijection between A and B. Along these lines, we have the following result which is again clear for finite sets A: Definition 32 N = ℵ 0 and R = c. The following is an axiom which cannot be proved or disproved. The Continuum Hypothesis There exists no set S such that ℵ 0 < S < c. If a set is infinite, we have mostly talked about the denumerable sets and the sets which are numerically equivalent to R. In case you think this is it for the infinite cardinals, we have the following result which is clear for finite sets: Theorem 20 For a set A, A < P(A). It turns out though that Theorem 21 The sets P(N) and R are numerically equivalent. deduce that 2 N and R are numerically equivalent. We may This last theorem and other results follow from the innocent looking but powerful result: Theorem 22 (The SchröderBernstein Theorem) If A and B are sets such that A B and B A then A = B. Warning: Note that although this looks like the simple result for x, y R, x y and y x implies that x = y, it is not true that infinite cardinalities are real numbers. This is actually stating that if there is a 11 function f : A B and a 11 function g : B A, then there exists a bijective function from A to B. Equivalently, if there exists a 11 function f : A B and a surjection h : A B then there exists a bijection between A and B. A useful lemma used in proving this theorem is 24
25 Lemma 2 If f : A B and g : C D are 11 (resp bijective) functions where A C = and B D =, then we can piece together a 11 (resp bijective) function h : A C B D with h A = f and h C = g. A curiosity is the following last observation (which we will not use and did not prove!!!) Theorem 23 The following statements are equivalent: 1. For 2 sets A and B, exactly one of the following occurs: A = B, A < B or A > B. 2. (The Axiom of Choice) For every collection of pairwise disjoint sets, there exists at least one set that contains exactly one element of each of these nonempty sets. 11 Chapter 11: Number Theory Many of the results in this Chapter explain things that we had hints about from Chapter 4 and 5. Definition 33 A prime is an integer p 2 whose only positive integer divisors are 1 and p. An integer that is not prime is called composite. Thus a number n 2 is composite if and only if n = ab for some 1 < a < n and 1 < b < n. Note that many results about divisibility were proved earlier in the course. For example, we proved 11.2 in Ch 4 (See section on divisibility of integers). Here s something we proved earlier but wasn t recorded above: Theorem 24 Let a, b Z be nonzero. 1. If a b and b a then a = ±b. 2. If a b then a b. We also proved the important Division Algorithm in order to prove that congruence modulo n 2 gives an equivalence relation on Z with distinct equivalence classes {[i] : 0 i n 1}. Theorem 25 (Division Algorithm: General Form) For integers a and b with a 0, there exist unique integers q and r such that b = aq+r and 0 r < a. 25
26 Note that the result for positive integers a and b is really just formalising long division of b by a where q is the unique quotient and r is the unique remainder which is naturally less than a and nonnegative. For general integers, it is easy to adjust the result from the positive integer case. Definition 34 For nonzero integers a, b, a common divisor c 0 is an integer with c a and c b. A greatest common divisor of nonzero integers a, b is d N such that d a and d b (i.e. d is a common divisor of a and b.) If d N is such that d a and d b then d d. Note: This is a slightly different definition from the one in the book but it is equivalent. First observe that the 2nd condition forces d to be larger than all other common divisors of a, b, since d d implies d d for positive integers. Second observe that greatest common divisors are unique by this definition: If we had 2 greatest common divisors of a, b, say d, d N then d d and d d and so d = d. We will then call the unique greatest common divisor of a and b, gcd(a, b). The Euclidean Algorithm below then guarantees the existence of gcd(a, b). Theorem 26 (Euclidean Algorithm) Let a, b Z be nonzero. Suppose 0 < a b. Then do a series of division algorithms as below: b = aq 0 + r 0, 0 r 0 < a (0) a = r 0 q 1 + r 1, 0 r 1 < r 0 (1) r 0 = r 1 q 2 + r 2, 0 r 2 < r 1 (2).. r k 2 = r k 1 q k + r k, 0 r k 1 < r k (k) r k 1 = r k q k+1, r k+1 = 0 (k + 1) If we assign also r 2 = b and r 1 = a, then the gcd of a and b exists and is computed as gcd(a, b) = r k where r k+1 = 0 is the first nonzero remainder above. This process must end as {r i } i 1 is a strictly decreasing sequence of positive integers less than or equal to a. 26
27 Note: gcd(a, b) = gcd( a, b ) = gcd( b, a ) for any nonzero integers a, b, so the Euclidean algorithm can always be used to find the gcd. Corollary 3 Let a, b Z be nonzero. There exist r, s Z such that ra + sb = gcd(a, b). Note that the Euclidean Algorithm can be used to recursively solve for r k = gcd(a, b) as an integer linear combination of a and b. Note also that the integers r and s above are not unique. The next few results are basically consequences of the Euclidean Algorithm and its corollary. Theorem 27 Let a, b, c Z, a, b nonzero. If a c and b c then It turns out that lcm(a, b) = ab. gcd(a,b) ab gcd(a, b) c Theorem 28 (Euclid s Lemma) Let a, b Z, where a 0. gcd(a, b) = 1 then a c. If a bc and Corollary 4 Let b, c Z and p a prime. If p bc then p b or p c. Corollary 5 Let a 1,..., a n Z and let p be prime. If p a 1 a n then p a i for some 1 i n. The existence of a prime factorisation was proven by strong induction earlier. The uniqueness of such a factorisation benefits from the corollary above of the Euclidean Algorithm. Theorem 29 (Fundamental Theorem of Arithmetic) Every integer n 2 can be written uniquely as a product of primes n = p 1 p k where p 1 p k are primes. Definition 35 The canonical factorisation of n 2 is n = p a 1 1 p a k k p 1 < < p k are distinct primes with multiplicities a i N. Corollary 6 Every integer greater than 1 has a prime factor where 27
28 Lemma 3 If n N is a composite number, then n has a prime factor p where p n. Note this lemma says that we can test whether a number n is prime by just checking that all the primes less than n are not divisors. The next two results clear up some questions posed in Chapter 5 about irrationality. Theorem 30 Let n N. Then n Q if and only if n Z. Corollary 7 If p is a prime, then p is irrational. The last result seems reasonable but requires proof: Theorem 31 The number of primes is infinite. 28
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