11 Sets II Operations
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1 11 Sets II Operations Tom Lewis Fall Term 2010 Tom Lewis () 11 Sets II Operations Fall Term / 12
2 Outline 1 Union and intersection 2 Set operations 3 The size of a union 4 Difference and symmetric difference 5 Cartesian products Tom Lewis () 11 Sets II Operations Fall Term / 12
3 Union and intersection Definition Let A and B be sets. Tom Lewis () 11 Sets II Operations Fall Term / 12
4 Union and intersection Definition Let A and B be sets. The union of A and B, denoted by A B, is the set of all elements that are in A or B (possibly both). Tom Lewis () 11 Sets II Operations Fall Term / 12
5 Union and intersection Definition Let A and B be sets. The union of A and B, denoted by A B, is the set of all elements that are in A or B (possibly both). The intersection of A and B, denoted by A B, is the set of all elements that are in A and B. Tom Lewis () 11 Sets II Operations Fall Term / 12
6 Union and intersection Problem Express A B and A B using set-builder notation and the logical symbols and. Tom Lewis () 11 Sets II Operations Fall Term / 12
7 Union and intersection Problem Express A B and A B using set-builder notation and the logical symbols and. Problem Express A B and A B using Venn diagrams. Tom Lewis () 11 Sets II Operations Fall Term / 12
8 Set operations Theorem Let A, B, and C be sets. The following are true: Tom Lewis () 11 Sets II Operations Fall Term / 12
9 Set operations Theorem Let A, B, and C be sets. The following are true: Commutative A B = B A and A B = B A Tom Lewis () 11 Sets II Operations Fall Term / 12
10 Set operations Theorem Let A, B, and C be sets. The following are true: Commutative A B = B A and A B = B A Associative A (B C) = (A B) C and A (B C) = (A B) C Tom Lewis () 11 Sets II Operations Fall Term / 12
11 Set operations Theorem Let A, B, and C be sets. The following are true: Commutative A B = B A and A B = B A Associative A (B C) = (A B) C and A (B C) = (A B) C Identity A = A and A = Tom Lewis () 11 Sets II Operations Fall Term / 12
12 Set operations Theorem Let A, B, and C be sets. The following are true: Commutative A B = B A and A B = B A Associative A (B C) = (A B) C and A (B C) = (A B) C Identity A = A and A = Distributive A (B C) = (A B) (A C) and A (B C) = (A B) (A C) Tom Lewis () 11 Sets II Operations Fall Term / 12
13 Set operations Theorem Let A, B, and C be sets. The following are true: Commutative A B = B A and A B = B A Associative A (B C) = (A B) C and A (B C) = (A B) C Identity A = A and A = Distributive A (B C) = (A B) (A C) and A (B C) = (A B) (A C) Problem Show that (A B) C = (C A) B. Tom Lewis () 11 Sets II Operations Fall Term / 12
14 Set operations Theorem Let A, B, and C be sets. The following are true: Commutative A B = B A and A B = B A Associative A (B C) = (A B) C and A (B C) = (A B) C Identity A = A and A = Distributive A (B C) = (A B) (A C) and A (B C) = (A B) (A C) Problem Show that (A B) C = (C A) B. Problem Prove the associative property for the union of sets. Tom Lewis () 11 Sets II Operations Fall Term / 12
15 The size of a union Definition (Disjoint) Tom Lewis () 11 Sets II Operations Fall Term / 12
16 The size of a union Definition (Disjoint) Let A and B be sets. We call A and B disjoint provided that A B =. Tom Lewis () 11 Sets II Operations Fall Term / 12
17 The size of a union Definition (Disjoint) Let A and B be sets. We call A and B disjoint provided that A B =. In general, let A 1, A 2,..., A n be a collection of sets. The sets are called pairwise disjoint provided that A i A j = whenever i j. In other words, no pair of sets has any elements in common. Tom Lewis () 11 Sets II Operations Fall Term / 12
18 The size of a union Definition (Disjoint) Let A and B be sets. We call A and B disjoint provided that A B =. In general, let A 1, A 2,..., A n be a collection of sets. The sets are called pairwise disjoint provided that A i A j = whenever i j. In other words, no pair of sets has any elements in common. Theorem (The addition principle) Let A 1, A 2,..., A n be pairwise disjoint. Then A 1 A 2... A n = A 1 + A A n. Tom Lewis () 11 Sets II Operations Fall Term / 12
19 The size of a union Problem There are 24 students taking Math 141, 19 students taking Bio 101, and 11 students taking both Math 141 and Bio 101. How many students are taking either course? Tom Lewis () 11 Sets II Operations Fall Term / 12
20 The size of a union Problem There are 24 students taking Math 141, 19 students taking Bio 101, and 11 students taking both Math 141 and Bio 101. How many students are taking either course? Theorem (Inclusion/exclusion principle) Let A and B be sets. Then A B = A + B A B. Tom Lewis () 11 Sets II Operations Fall Term / 12
21 The size of a union Problem A keypad contains the digits 0 through 9. An access code consists of selecting 4 keys in succession, repetitions allowed. How many codes begin with a 3 or end with an 8? Tom Lewis () 11 Sets II Operations Fall Term / 12
22 The size of a union Problem A keypad contains the digits 0 through 9. An access code consists of selecting 4 keys in succession, repetitions allowed. How many codes begin with a 3 or end with an 8? Problem How many integers between 1 and 100 (inclusive) are divisible by 2 or 5? Tom Lewis () 11 Sets II Operations Fall Term / 12
23 Difference and symmetric difference Definition (Set difference) Let A and B be sets. The difference between A and B, denoted by A B, is the set of all elements of A that are not in B. A B = {x : x A and x / B} Tom Lewis () 11 Sets II Operations Fall Term / 12
24 Difference and symmetric difference Definition (Set difference) Let A and B be sets. The difference between A and B, denoted by A B, is the set of all elements of A that are not in B. Definition (Symmetric difference) A B = {x : x A and x / B} Let A and B be sets. The symmetric difference between A and B, denoted by A B, is the set of all elements in A but not B or in B but not A. A B = (A B) (B A). Tom Lewis () 11 Sets II Operations Fall Term / 12
25 Difference and symmetric difference Theorem (DeMorgan s law) Let A, B, and C be sets. Then A (B C) = (A B) (A C) and A (B C) = (A B) (A C) Tom Lewis () 11 Sets II Operations Fall Term / 12
26 Difference and symmetric difference Theorem (DeMorgan s law) Let A, B, and C be sets. Then A (B C) = (A B) (A C) and A (B C) = (A B) (A C) Proof. The proof is an exercise. Do these make sense by Venn diagrams? Tom Lewis () 11 Sets II Operations Fall Term / 12
27 Cartesian products Problem An experiment consists of tossing a red die and a green die. Describe the set of outcomes of this experiment. Tom Lewis () 11 Sets II Operations Fall Term / 12
28 Cartesian products Problem An experiment consists of tossing a red die and a green die. Describe the set of outcomes of this experiment. Definition Let A and B be sets. The Cartesian product of A and B, denoted by A B, is the set of all possible ordered pairs (a, b) where a A and b B. That is, A B = {(a, b) : a A, b B} Tom Lewis () 11 Sets II Operations Fall Term / 12
29 Cartesian products Problem An experiment consists of tossing a red die and a green die. Describe the set of outcomes of this experiment. Definition Let A and B be sets. The Cartesian product of A and B, denoted by A B, is the set of all possible ordered pairs (a, b) where a A and b B. That is, A B = {(a, b) : a A, b B} Problem Construct an explicit example to show that A B is not necessarily equal to B A. Tom Lewis () 11 Sets II Operations Fall Term / 12
30 Cartesian products Theorem Let A and B be sets. Then A B = A B. Tom Lewis () 11 Sets II Operations Fall Term / 12
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