Sets. Definition A set is an unordered collection of objects called elements or members of the set.

Transcription

1 Sets Definition A set is an unordered collection of objects called elements or members of the set.

2 Sets Definition A set is an unordered collection of objects called elements or members of the set. Examples: S = {1, 2, 5} V = {a, e, i, o, u} X = {apple, 32, A, Z} Y = {2, 3, 4, 6, 22}

3 Sets Definition A set is an unordered collection of objects called elements or members of the set. Examples: S = {1, 2, 5} V = {a, e, i, o, u} X = {apple, 32, A, Z} Y = {2, 3, 4, 6, 22} Y = {2, 3, 3, 4, 3, 22, 2, 6} Y = {22, 4, 6, 22, 3}

4 Sets Definition A set is an unordered collection of objects called elements or members of the set. Examples: S = {1, 2, 5} V = {a, e, i, o, u} X = {apple, 32, A, Z} Y = {2, 3, 4, 6, 22} Y = {2, 3, 3, 4, 3, 22, 2, 6} Y = {22, 4, 6, 22, 3} Familiar sets: N, Z, R, Q, R +.

5 Sets Definition A set is an unordered collection of objects called elements or members of the set. Examples: S = {1, 2, 5} V = {a, e, i, o, u} X = {apple, 32, A, Z} Y = {2, 3, 4, 6, 22} Y = {2, 3, 3, 4, 3, 22, 2, 6} Y = {22, 4, 6, 22, 3} Familiar sets: N, Z, R, Q, R +. Two common questions does x S? is X = Y?

6 Sets Definition A set is an unordered collection of objects called elements or members of the set. Examples: S = {1, 2, 5} V = {a, e, i, o, u} X = {apple, 32, A, Z} Y = {2, 3, 4, 6, 22} Y = {2, 3, 3, 4, 3, 22, 2, 6} Y = {22, 4, 6, 22, 3} Familiar sets: N, Z, R, Q, R +. Two common questions does x S? is X = Y? Roster method vs set-builder method.

7 Equality and containment A = B Y = {2, 3, 4, 6, 22}

8 Equality and containment A = B Y = {2, 3, 4, 6, 22} Y = {2, 3, 4, 3, 22, 2, 6} Y = {22, 4, 6, 22, 3}

9 Equality and containment A = B Y = {2, 3, 4, 6, 22} Y = {2, 3, 4, 3, 22, 2, 6} Y = {22, 4, 6, 22, 3} Express equality using quantifiers.

10 Equality and containment A = B Y = {2, 3, 4, 6, 22} Y = {2, 3, 4, 3, 22, 2, 6} Y = {22, 4, 6, 22, 3} Express equality using quantifiers. x (x A x B)

11 Equality and containment A = B A B Y = {2, 3, 4, 6, 22} Y = {2, 3, 4, 3, 22, 2, 6} Y = {22, 4, 6, 22, 3} Express equality using quantifiers. A = {a, b, 22, 33} B = {a, b, 22, 33, N} x (x A x B)

12 Equality and containment A = B A B Y = {2, 3, 4, 6, 22} Y = {2, 3, 4, 3, 22, 2, 6} Y = {22, 4, 6, 22, 3} Express equality using quantifiers. A = {a, b, 22, 33} B = {a, b, 22, 33, N} B A Express not a subset of using quantifiers. x (x A x B)

13 Equality and containment A = B A B Y = {2, 3, 4, 6, 22} Y = {2, 3, 4, 3, 22, 2, 6} Y = {22, 4, 6, 22, 3} Express equality using quantifiers. x (x A x B) A = {a, b, 22, 33} B = {a, b, 22, 33, N} B A Express not a subset of using quantifiers. x (x B x / A)

14 Empty set Definition A set that has no elements is the empty set denoted by φ or {}.

15 Empty set Definition A set that has no elements is the empty set denoted by φ or {}. for any set S, φ S.

16 Empty set Definition A set that has no elements is the empty set denoted by φ or {}. for any set S, φ S. for any set S, φ S.

17 Empty set Definition A set that has no elements is the empty set denoted by φ or {}. for any set S, φ S. for any set S, φ S. proof?

18 Empty set Definition A set that has no elements is the empty set denoted by φ or {}. for any set S, φ S. for any set S, φ S. proof? x (x φ x S) for any x, x φ always evaluates to false. the conditional (x φ x S) evaluates to true.

19 Empty set Definition A set that has no elements is the empty set denoted by φ or {}. for any set S, φ S. for any set S, φ S. proof? x (x φ x S) for any x, x φ always evaluates to false. the conditional (x φ x S) evaluates to true. Example of a vacuous proof.

20 Empty set Definition A set that has no elements is the empty set denoted by φ or {}. for any set S, φ S. for any set S, φ S. proof? x (x φ x S) for any x, x φ always evaluates to false. the conditional (x φ x S) evaluates to true. Example of a vacuous proof. Important {φ} is not the same as φ.

21 Power set S = {32, 6, 41} subsets of S: {}, {32}, {6}, {41}, {32, 6}, {6, 41}, {32, 41}, {32, 6, 41}.

22 Power set S = {32, 6, 41} subsets of S: {}, {32}, {6}, {41}, {32, 6}, {6, 41}, {32, 41}, {32, 6, 41}. Definition The set of all subsets of S is called the power set of S, denoted as P(S).

23 Power set S = {32, 6, 41} subsets of S: {}, {32}, {6}, {41}, {32, 6}, {6, 41}, {32, 41}, {32, 6, 41}. Definition The set of all subsets of S is called the power set of S, denoted as P(S). The power set of a set with n elements has 2 n elements.

24 Size of a set Definition If there are exactly n distinct elements in a set S, where n is a non-negative integer, then we say S is finite and cardinality of S is n. S = {1, 17, a, b}; S = 4. S = {1, 2, 2, 6, 7, 7}; S = 4.

25 Size of a set Definition If there are exactly n distinct elements in a set S, where n is a non-negative integer, then we say S is finite and cardinality of S is n. S = {1, 17, a, b}; S = 4. S = {1, 2, 2, 6, 7, 7}; S = 4. φ = 0.

26 Size of a set Definition If there are exactly n distinct elements in a set S, where n is a non-negative integer, then we say S is finite and cardinality of S is n. S = {1, 17, a, b}; S = 4. S = {1, 2, 2, 6, 7, 7}; S = 4. φ = 0. A set that is not finite is said to be infinite

27 Set operations Union Intersection Set difference Exclusive OR Complement Set identities

28 Exercises 1. For each of the sets below determine if 2 belongs to the set. (a) {x R x is an integer greater than 1} (b) {x R x is the square of an integer} (c) {2, {2}} (d) {{2}, {2}} (e) {{2}, {2, {2}}} 2. Let A = {φ, b}. Construct the following sets: A φ {φ} A A P(A) A P(A) 3. Prove that P(A) P(B) iff A B. 4. Express A B using quantifiers. 5. Prove that A B = Ā B.