L01  Set Theory. CSci/Math May 2015


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1 L01  Set Theory CSci/Math May / 10
2 Elements and Subsets of a Set a A means a is an element of the set A a / A means a is not an element of the set A A B means A is a subset of B (all elements of A are also in B) A B means A is not a subset of B (at least one element of A is not in B) A B means A B and A B 2 / 10
3 Elements and Subsets of a Set a A means a is an element of the set A a / A means a is not an element of the set A A B means A is a subset of B (all elements of A are also in B) A B means A is not a subset of B (at least one element of A is not in B) A B means A B and A B Example 1 A = {2, 3, 4, {5}} Which of the following are elements of A? Which are subsets? (a) 2 (b) {2} (c) 5 (d) {5} (e) {2, 3, 4} (f) {2, 3, 4, {5}} (g) {{5}} (h) {{{5}}} 2 / 10
4 Set Builder Notation A = {expression : rule} = {expression rule} Example 2 A = { 1 n n N} B = { 1 n n N, 1 n 5} C = {x x B, x < 1} D = {2, 4, 6, 8, 10, 12} E = {1, 3, 5, 7, 9,...} List the elements in A, B, and C. Write D and E in set builder notation. What are C and D? 3 / 10
5 Ordered Pairs and Cartesian Product Example 3 Your society is electing a new executive. 4 / 10
6 Ordered Pairs and Cartesian Product Example 3 Your society is electing a new executive. The ordered pair (x, y) indicates that x is president and y is vicepresident. Is (Alice, Bob)=(Bob, Alice)? 4 / 10
7 Ordered Pairs and Cartesian Product Example 3 Your society is electing a new executive. The ordered pair (x, y) indicates that x is president and y is vicepresident. Is (Alice, Bob)=(Bob, Alice)? Let A = {people running for president} and B = {people running for vicepresident}. What does A B mean? 4 / 10
8 Ordered Pairs and Cartesian Product Example 3 Your society is electing a new executive. The ordered pair (x, y) indicates that x is president and y is vicepresident. Is (Alice, Bob)=(Bob, Alice)? Let A = {people running for president} and B = {people running for vicepresident}. What does A B mean? If A = {Alice, Chris, Dean} and B = {Bob, Eve, Fiona, Georgie}, what is A B, and what does this number mean? 4 / 10
9 Ordered Pairs and Cartesian Product Example 3 Your society is electing a new executive. The ordered pair (x, y) indicates that x is president and y is vicepresident. Is (Alice, Bob)=(Bob, Alice)? Let A = {people running for president} and B = {people running for vicepresident}. What does A B mean? If A = {Alice, Chris, Dean} and B = {Bob, Eve, Fiona, Georgie}, what is A B, and what does this number mean? If A = {candidates for president}, B = {candidates for vicepresident}, C = {candidates for secretary}, D = {candidates for treasurer}, E = {candidates for DSU rep}, what is A B C D E? 4 / 10
10 Union, Intersection, Difference, Complement A B means the union of A and B, i.e. the set consisting of all element in A and B A B means the intersection of A and B, i.e. the set consisting of all elements which are both in A and B A \ B = A B means the difference of A and B, i.e. the set of elements in A which are not in B A c = A means the complement of A, i.e. U \ A where U is the universal set 5 / 10
11 Union, Intersection, Difference, Complement Example 4 Let A = {a, b, c, d, e}, B = {a, d, e}, C = {a, b, c, f, g, h}, and U = {a, b, c, d, e, f, g, h, i, j}. Find (1) A B (2) A B (3) A C (4) A C (5) A B (6) B A (7) A C (8) C (A B) (9) A (10) C (11) A B (12) A C 6 / 10
12 Union, Intersection, Difference, Complement Example 5 A = {2n n Z} {3m m Z} (1) Which of the following are elements of A? (a) 4 (b) 9 (c) 6 (d) 90 (e) 15 (2) Which of the following are subsets of A? (a) {6, 12, 18} (b) {6, 9, 12, 15} (c) { 6, 90, 12} 7 / 10
13 Union, Intersection, Difference, Complement Example 6 A (B C) (1) Draw the Venn Diagram. (2) Can you find another way using only unions and intersections to describe this set? 8 / 10
14 Union, Intersection, Difference, Complement Example 6 A (B C) (1) Draw the Venn Diagram. (2) Can you find another way using only unions and intersections to describe this set? This is one half of the distributive laws. They are A (B C) = (A B) (A C), A (B C) = (A B) (A C). 8 / 10
15 Union, Intersection, Difference, Complement Example 7 (A B) (1) Draw the Venn Diagram. (2) Can you find another way using only complements, unions and intersections to describe this set? 9 / 10
16 Union, Intersection, Difference, Complement Example 7 (A B) (1) Draw the Venn Diagram. (2) Can you find another way using only complements, unions and intersections to describe this set? This is one half of DeMorgan s laws. They are (A B) = A B, (A B) = A B. 9 / 10
17 Indexed Sets Example 8 U = {all living people} A i = {x x was born on the ith day of the month} B j = {x x was born during the jth month of the year} C k = {x x was born on the kth day of the week} Describe in ( words: 3 ) (a) B 1 C k k=1 (c) C 6 (e) 1 j 4 B j 5 (b) (d) ( 15 B j j=1 i=12 1 k 5 C k A i ) 1 j 4 B j 10 / 10
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