# Logic & Discrete Math in Software Engineering (CAS 701) Dr. Borzoo Bonakdarpour

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1 Logic & Discrete Math in Software Engineering (CAS 701) Background Dr. Borzoo Bonakdarpour Department of Computing and Software McMaster University Dr. Borzoo Bonakdarpour Logic & Discrete Math in SE (CAS 701) - McMaster University 1/26

2 Presentation outline Sets Relations Functions Set Cardinality Induction and Proofs Dr. Borzoo Bonakdarpour Logic & Discrete Math in SE (CAS 701) - McMaster University 2/26

3 Sets A set is a collection of objects called members or elements. We write α S to mean that α is a member of S (α S is the opposite). We write α 1,..., α n S to mean that α 1 S,..., and α n S. Dr. Borzoo Bonakdarpour Logic & Discrete Math in SE (CAS 701) - McMaster University 3/26

4 Sets Two sets are equal (i.e., S = T ) iff they have the same members: for every x, x S iff x T. S is said to be a subset of T (i.e., S T ) iff for every x, x S implies x T. Every set is a subset of itself. S = T iff S T and T S. Dr. Borzoo Bonakdarpour Logic & Discrete Math in SE (CAS 701) - McMaster University 4/26

5 Sets S is a proper subset of T (i.e., S T ), iff S T and S T. Sets are not ordered (e.g., {α, β} = {β, α}). Repetition in sets is not important (e.g., {α, α, β} = {α, β}). The empty set has no members. Hence, S for all S (why?). Dr. Borzoo Bonakdarpour Logic & Discrete Math in SE (CAS 701) - McMaster University 5/26

6 Sets What are the concrete set that represent: {x x < 100 and x is prime} {x x = 0 or x = 1 or x = 2} Dr. Borzoo Bonakdarpour Logic & Discrete Math in SE (CAS 701) - McMaster University 6/26

7 Sets We define S = {x x S} (complement) S T = {x x S or x T } (union) S T = {x x S and x T } (intersection) S T = {x x S and x T } (difference) Dr. Borzoo Bonakdarpour Logic & Discrete Math in SE (CAS 701) - McMaster University 7/26

8 Sets We define S i = {x x S i for some i I} i I S i = {x x S i for all i I} i I Dr. Borzoo Bonakdarpour Logic & Discrete Math in SE (CAS 701) - McMaster University 8/26

9 Presentation outline Sets Relations Functions Set Cardinality Induction and Proofs Dr. Borzoo Bonakdarpour Logic & Discrete Math in SE (CAS 701) - McMaster University 9/26

10 Relations The ordered pair of objects α and β is written as α, β. Then α, β = α 1, β 1 iff α = α 1 and β = β 1. Similarly, one can define and ordered n-tuple α 1,..., a n. One can also define a set of ordered pairs (e.g., { m, n m, n are natural numbers and m < n}). Dr. Borzoo Bonakdarpour Logic & Discrete Math in SE (CAS 701) - McMaster University 10/26

11 Relations The Cartesian product of sets S 1,..., S n is defined by S 1 S n = { x 1,..., x n x 1 S 1,..., x n S n} Let S n = S S }{{} n An n-ary relation R on set S is a subset of S n. A special binary relation is the equality relation: or { x, y x, y S and x = y} { x, x x S} Dr. Borzoo Bonakdarpour Logic & Discrete Math in SE (CAS 701) - McMaster University 11/26

12 Relations For a binary relation R, we often write xry to denote x, y R. R is reflexive on S, iff for any x S, xrx. R is symmetric on S, iff for any x, y S, whenever xry, then yrx. R is transitive on S, iff for any x, y, z S, whenever xry and yrz, then xrz. R is an equivalence relation iff R is reflexive, symmetric, and transitive. Dr. Borzoo Bonakdarpour Logic & Discrete Math in SE (CAS 701) - McMaster University 12/26

13 Relations Suppose R is an equivalence relation on S. For any x S the set x = {y S xry} is called the R-equivalence class of x. R-equivalence classes make a partition of S. The restriction of R to S 1 is the n-ary relation R S1. n Dr. Borzoo Bonakdarpour Logic & Discrete Math in SE (CAS 701) - McMaster University 13/26

14 Presentation outline Sets Relations Functions Set Cardinality Induction and Proofs Dr. Borzoo Bonakdarpour Logic & Discrete Math in SE (CAS 701) - McMaster University 14/26

15 Functions A function (mapping) f is a set of ordered pairs such that if x, y f and x, z f, then y = z. The domain dom(f ) of f is the set {x x, y f for some y} The range ran(f ) of f is the set {y x, y f for some x} Dr. Borzoo Bonakdarpour Logic & Discrete Math in SE (CAS 701) - McMaster University 15/26

16 Functions f (x) denotes the unique element in y ran(f ), where x dom(f ) and x, y f. If f is a function with dom(f ) = S and ran(f ) T, we say that f is a function from S to T and denote it by f : S T Similarly, one can define n-ary functions. Suppose f : S T is a function and S 1 S. The restriction of f to S 1 is the function f S 1 : S 1 T Dr. Borzoo Bonakdarpour Logic & Discrete Math in SE (CAS 701) - McMaster University 16/26

17 Functions A function f : S T is onto if ran(f ) = T A function is one-to-one if f (x) = f (y) implies x = y. Dr. Borzoo Bonakdarpour Logic & Discrete Math in SE (CAS 701) - McMaster University 17/26

18 Presentation outline Sets Relations Functions Set Cardinality Induction and Proofs Dr. Borzoo Bonakdarpour Logic & Discrete Math in SE (CAS 701) - McMaster University 18/26

19 Set Cardinality Two sets S and T are equipotent (i.e., S T ) iff there is a one-to-one mapping from S onto T. is an equivalence relations (why?) A cardinal of a set S is denoted by S where: S = T iff S T. A set S is to be countably infinite, iff S = N. A set S is said to be countable, iff S N (i.e., S is finite or countably infinite). Dr. Borzoo Bonakdarpour Logic & Discrete Math in SE (CAS 701) - McMaster University 19/26

20 Set Cardinality Theorem 1. A subset of a countable set is countable. Theorem 2. The union of any finite number of countable sets is countable. Theorem 3. The union of any countably many countable sets is countable. Dr. Borzoo Bonakdarpour Logic & Discrete Math in SE (CAS 701) - McMaster University 20/26

21 Set Cardinality Theorem 4. The Cartesian product of any finite number of countable sets is countable. Theorem 5. The set of all finite sequences with the members of a countable set as components is countable. Dr. Borzoo Bonakdarpour Logic & Discrete Math in SE (CAS 701) - McMaster University 21/26

22 Presentation outline Sets Relations Functions Set Cardinality Induction and Proofs Dr. Borzoo Bonakdarpour Logic & Discrete Math in SE (CAS 701) - McMaster University 22/26

23 Induction and Proofs Inductive definition of natural numbers: [1 ] 0 N. [2 ] For any n, if n N, then n N, where n is the successor of n. [3 ] n N only if n has been generated by [1] and [2]. Dr. Borzoo Bonakdarpour Logic & Discrete Math in SE (CAS 701) - McMaster University 23/26

24 Induction and Proofs Another inductive definition of natural numbers: [1 ] 0 S. [2 ] For any n, if n S, then n S Dr. Borzoo Bonakdarpour Logic & Discrete Math in SE (CAS 701) - McMaster University 24/26

25 Induction and Proofs Theorem. Suppose R is a unary relation. If [1 ] R(0). [2 ] For any n N, if R(n), then R(n ). then R(n) for any n N. Dr. Borzoo Bonakdarpour Logic & Discrete Math in SE (CAS 701) - McMaster University 25/26

26 Example Prove that n = n(n + 1) 2 Dr. Borzoo Bonakdarpour Logic & Discrete Math in SE (CAS 701) - McMaster University 26/26

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