Lecture 4 -- Sets, Relations, Functions 1

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1 Lecture 4 Sets, Relations, Functions Pat Place Carnegie Mellon University Models of Software Systems Fall 1999 Lecture 4 -- Sets, Relations, Functions 1

2 The Story So Far Formal Systems > Syntax» language: alphabet + grammar» deductive system: axioms + rules of inference > Semantics» interpretation Propositional logic Predicate logic (with equality) Natural deduction Equational reasoning New theories Lecture 4 -- Sets, Relations, Functions 2

3 This Lecture We will develop one of the most important theories for modeling software systems: set theory (including relations and functions) Lecture 4 -- Sets, Relations, Functions 3

4 Sets Collection of distinct objects In this course elements of wellformed sets are drawn from a larger homogeneous set Examples: > {1, 3, 5, 7,... } > {red, green, blue, green} > {Steve, John, Angel, Asundi,... } > { yes, no } But not: {red, 1, 2, 3 }?=? {blue, red, green} Lecture 4 -- Sets, Relations, Functions 4

5 Set Definition & Membership Sets can be defined by enumeration ( in extension ) Odds == {1, 3, 5, 7,... } Colors == {red, green, blue, blue, green} == {Steve, John, Angel, Asundi,... } > Note the use of == for definition ( ) Membership test: 3 Odds ; 2 Odds > note is a predicate over sets and elements Lecture 4 -- Sets, Relations, Functions 5

6 Set Names Some sets have predefined names the set with no elements (the null or empty set) -- also written {} the set of natural numbers {0, 1,... } the set of integers {... -2, -1, 0, 1, 2,... } 1 the set of natural numbers (except 0) {1, 2, } Lecture 4 -- Sets, Relations, Functions 6

7 Given Sets We can define new sets of primitive elements > Called basic sets > Written [TheSet] > We know nothing about the elements, but assume there is an = operator over them Examples > [BookIdentifiers] > [Date,Name,Place] Later we will see how we can add assertions about the elements of a given set Lecture 4 -- Sets, Relations, Functions 7

8 Set Equality and Cardinality Sets have an equality operator Two sets are equal if they have the same elements. That is: (S = T) ( x x S x T) Note this implies that ordering of elements does not matter Cardinality (size) of a set: # > # {1, 2, 4} = 3 > # { {1,2}, {1,2,3,4,5,6,7} } =? > # {1, 2, 2, 4} =? Qn: Why? Lecture 4 -- Sets, Relations, Functions 8

9 Set Operators Can form new sets from other sets using operators: > (intersection), (union), \ (difference) Examples: Let A == {1,2,3} and B == {3,4,5} > A B = {3} > A B = {1,2,3,4,5} > A \ B = {1,2} Note the use of "=" versus "==" Qn: Why? Lecture 4 -- Sets, Relations, Functions 9

10 Subsets A is a subset of B ( A B) if every element of A is also an element of B A = B if A B and B A Qn: Why? Lecture 4 -- Sets, Relations, Functions 10

11 Set Axioms & Laws Basic axioms > Set membership: x x {x} > Empty set: x (x ) Laws (can be proven) > A B = B A > A B = B A > (A B) C = A (B C) > (A B) C = A (B C) > A (B C) = (A B ) (A C ) > and lots of others Qn: How would you prove these laws? Lecture 4 -- Sets, Relations, Functions 11

12 Example Prove A B = B A (hint use an equational style) Note need formal def of intersection before you can do this Def: x (x A B x A x B) A B = x x A x B -definition = x x B x A -commutativity = B A -commutativity Lecture 4 -- Sets, Relations, Functions 12

13 Set Comprehension Enumerating all of the elements of a set is not always possible Would like to describe a set by in terms of a distinguishing property of its elements. > Roster == the set of students in > Pgh == the set of residents of Pittsburgh > Primes == the set of integers that are prime Lecture 4 -- Sets, Relations, Functions 13

14 Set Comprehension (2) Each element "satisfies"some "criterion" Qn: How can we define such criteria? Ans: Predicates! This kind of set specification is called set comprehension Lecture 4 -- Sets, Relations, Functions 14

15 Set Comprehension (2) Simple form of set comprehension {x : S P(x)} the set of all x in S that satisfy P(x), or the set of all x in S such that P(x) Examples > natural numbers less than 20: {x: x < 20} > even integers: {x: ( y: x = 2y)}» note we identify the types of variables in predicates > all natural numbers: {x: true} > empty set of natural numbers: {x: false} Lecture 4 -- Sets, Relations, Functions 15

16 Set Comprehension (3) Sometimes it is helpful to specify a pattern for the elements > We will use the form: {x: S f(x)}, where f is some function defined on elements of S > Note the use of instead of Lecture 4 -- Sets, Relations, Functions 16

17 Set Comprehension (4) Examples: > squares: {x: x 2 } > authors of books in library: Qn: How might this have been written using just predicates? {b: Library author-of(b) } where author-of: Book Author > birthdays of people in {person: birthday-of(person) } Lecture 4 -- Sets, Relations, Functions 17

18 Alternative Notation In more traditional mathematics the expression part is usually written first Examples > {x: x 2 } would be written { x 2 x } For reasons that will be come clear later in the course we do not do this Lecture 4 -- Sets, Relations, Functions 18

19 Set Comprehension (5) Most general form combines the two forms {x: T P(x) f(x)} Examples > squares of natural numbers less than 20: {x: x < 20 x 2 } > squares of even integers: {x: ( y: x = 2y) x 2 } Can think of this as nested pipelines S P (filter) F (computation) Lecture 4 -- Sets, Relations, Functions 19

20 Power Sets The set of all subsets of S is referred to as the power set of S and written S Examples: {1, 2, 3 } = {, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}} = {, {1},{1,2}, {1,3}...} Lecture 4 -- Sets, Relations, Functions 20

21 Power Sets (2) Power sets can be used to define new types, and can be used in declarations > class-groups == Student > Integer-sets == > If x :, then x is a set of Integers. > Whenever you see.. :.. read set of > Alternatively could write: x: Integer-sets Note difference between : and Lecture 4 -- Sets, Relations, Functions 21

22 Cartesian Products Ordered pairs, triples, etc. Examples: > (2, 3) (3, 2) > S == { (2, red), (5, blue), (3, red) } The set of all tuples constructed from two sets is called a Cartesian Product, or just Product of those sets > If S and T are those sets, this is written S T Lecture 4 -- Sets, Relations, Functions 22

23 Cartesian Products (2) Examples pairs of natural numbers > (2, red): Color > S: ( Color) Lecture 4 -- Sets, Relations, Functions 23

24 More Tuples In general tuples can be described using variables, much like the record constructor of Pascal General form: > {declarations predicate} > the predicate is sometimes called an invariant over the state space defined by the declarations Lecture 4 -- Sets, Relations, Functions 24

25 More Tuples (2) Examples > {(x, y): y = x + 1} = {(0,1), (1,2), (2,3), } > {x, y: y = x + 1} = {(0,1), (1,2), (2,3), } > {x: ; y: y = #x} = { ({-1,2,3}, 3), (, 0) } Lecture 4 -- Sets, Relations, Functions 25

26 Relations A relation is a set of pairs Examples: > A == { (1,1), (1,2), (2,2) } > B == { (2, red), (5, blue), (3, red) } > C == { (David, Jun 1}, (Mary, Aug 2), (Bill, Feb 5) } Lecture 4 -- Sets, Relations, Functions 26

27 Relations (2) The set of all relations over sets S, T is indicated by S T > In A: T 1 T 2 we call T 1 the source and T 2 the target > S T is equivalent to (S x T) Examples: A: B: Color C: Person Date If S has 3 elements and T has 2 elements how many does S T have? Lecture 4 -- Sets, Relations, Functions 27

28 Relations (3) The domain of a relation is the set of first elements ( dom ) The range of a relation is the set of second elements ( ran ) Lecture 4 -- Sets, Relations, Functions 28

29 Relations (4) Examples: A == { (1,1), (1,2), (2,2) } dom A = { 1, 2 } and ran A = { 1, 2 } even though A B == { (2, red), (5, blue), (3, red) } dom B = { 2,3,5 } and ran B = { red, blue } C == { (David, Jun 1), (Mary, Aug 2), (Bill, Feb 5) } dom C = { David, Mary, Bill} ran C = { Jun 1, Aug 2, Feb 5 } Lecture 4 -- Sets, Relations, Functions 29

30 Functions A function is a relation such that no two distinct pairs contain the same first element Examples: B == { (2, red), (5, blue), (3, red) } C == { (David, Jun 1), (Mary, Aug 2}, {Bill, Feb 5) } No: A == { (1,1), (1,2), (2,2) } Lecture 4 -- Sets, Relations, Functions 30

31 Some special cases Suppose f: A B 1. f is a function defined for all values of A we say f is a total function, and write A B 2. f is a function defined for some values of A we say f is a partial function, and write A B 3. f is a function defined for a finite set of values of A we say f is a finite function, and write A B Lecture 4 -- Sets, Relations, Functions 31

32 Some special cases (2) 4. f is a function for which no element in ran(f) is associated with more than one element in dom(f) we say f is a one-to-one or injective function, and write A B (total) A B (partial) 5. f is a function whose range is B we say f is an onto or surjective function, and write A B (total) A B (partial) 6. f is both one-to-one and onto we say f is a bijection, and write A B (total) Lecture 4 -- Sets, Relations, Functions 32

33 Some special cases (3) relations partial functions injective total functions bijective surjective Lecture 4 -- Sets, Relations, Functions 33

34 Relations/Functions as Sets Since relations are just sets (of pairs) we can apply set operators to them. Example: > R1 = {(1,red), (2,blue)} > R2 = {(3,green), (2,blue)} > R1 R2 = {(1,red), (2,blue), (3,green)} > # R1 = 2 Question: is the union/intersection of two functions a function? Question: how would you prove it? Lecture 4 -- Sets, Relations, Functions 34

35 Relational Composition If the range of one relation is the domain of another can form the composition (R1 R2) A 0 B 1 X 2 Y R1 3 R2 red blue green Note: in some texts ran(r2) must be the same as dom(r2) for to be defined Lecture 4 -- Sets, Relations, Functions 35

36 Overwriting Frequently we will want to change the value of a function for one or more values The overwriting operator does this: Example: > f == {(1,red), (2,blue), (3,green)} > g == {(1,pink), (4,mauve)} > f g = {(1,pink), (2,blue), (3,green), (4,mauve) } Note: replacement only considers domain values Lecture 4 -- Sets, Relations, Functions 36

37 Other Operators There is a rich library of other auxiliary definitions. We ll be encountering many as we proceed through the class Some examples: >, the set of natural numbers > 1, the set of positive natural numbers >.., the "between" operator: 2..5= {2,3,4,5} Lecture 4 -- Sets, Relations, Functions 37

38 Axiomatic Declarations Formally when we want to declare new variables we use an "axiomatic schema" It looks like this f: x: f(x) = 3x S: S = {1, 3, 8} The final definition is equivalent to S == {1, 3, 8} Lecture 4 -- Sets, Relations, Functions 38

39 Summary of Set Building Operators Suppose S, R, T are sets S R (intersection), (union), \ (difference) S (power set) {x: S P(x)} (set comprehension} {x: S Exp(x)} {x: S P(x) Exp(x)} (generalized set comprehension) S R (cartesian product) S R (relations) S R (functions), etc. (relational composition) (overwriting) Lecture 4 -- Sets, Relations, Functions 39

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