CS1802 Week 6: Sets Operations, Product Sum Rule Pigeon Hole Principle (Ch )

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1 CS1802 Discrete Structures Recitation Fall 2017 October 9-12, 2017 CS1802 Week 6: Sets Operations, Product Sum Rule Pigeon Hole Principle (Ch ) Sets i. Set Notation: Draw an arrow from the box on the left to the box on the right to match notation or phrases to the equivalent definition. 1. Notation Definition A B Intersection A B Subset and not equal (i.e. a proper subset such that A is inside B but not equal. A has less elements) A B The complement A B Union A B Subset and may be equal (i.e. A = B) Ā The union of A and B without the intersection of A and B (i.e.a B disjoint (Also a gate we have learned about)) The empty set ii. Express the following in English. Then write the equivalent set expressed and its cardinality. 1. x = 5 : {x, x 2, x 3, x 4 } 2. {x x Z and x < 4} 1

2 iii. Compute the following set given: A = {1, 2, 3, 4, 5}, B = { 1, 2, 3, 4, 5}, C = { 4, 2, 0, 2, 4}. Draw a Venn Diagram to visually represent each resulting set (Note you do not need to add in the numbers). Let U = set of all integers 1. A B 2. A B 3. (A B) C 4. C U iv. Compute the Cartesian product of the given sets to generate all of the ordered tuples. A = {a, b, c}, B = {1, 2, 3}, C = {6}. 1. A B 2. A C v. Compute the power sets below given the following. X = {1, 2, 3}, Y = { }. Note to yourself, if given a set of 3 elements, how many subsets total will you have in your power set? 1. P(X) 2. P(Y ) 2

3 vi. Given the following Set, compute the computer representation(i.e. a bit-string) using 0s and 1s. 1. A = {0, 1, 2, 3, 4, 5, 7} 2. Ā = {6} 3. B = {0, 2, 4, 6} 4. B {1, 3, 5} Set Operations i. Given U = {pink, red, blue, white, yellow, black} A = {red, blue, yellow, black} B= {pink, red, white, yellow} C = {white} Determine: A-B B-A A B B-A-C P(A) {V P(A): V = 1} 3

4 Cross Product Operation i. Given the sets defined above, determine A B and draw the results in a color x color 2 dimensional coordinate system Sets as bit strings i. Represent U, A, B, C, A-B, B-A, B-A-C, A B from problem 1 as a bit string. 4

5 Proof with sets i. Prove that { x Z: 26 x} { x Z: 13 x} Counting i. Greta plans to register for one course at MyWay University. The school has 4 departments: English, Math, History and Science. The English department offers 7 courses Greta is eligible to take, the Math department offers 3 courses Greta is eligible to take, the History department offers 5 courses Greta is eligible to take and the Science department offers 9 courses Greta is eligible to take. How many different choices does Greta have for her course? ii. Bob is building a garden. He has dug out 6 different plots. He arrives at the garden center to see that the store is offering seven different perennials for sale. How many different perennial garden configurations are available for Bob s garden? iii. How many positive integers less than 111 are divisible by 11 or 2 or 5? Basic Set Operations Consider subsets A = {a, f, d}, B = {b, d, h, e} of the universal set U = {a, b, c, d, e, f, g, h}. Compute each of the following. 1. A B 2. A B 3. B A 4. B 5. A B 6. {a, b} {b, c} 7. (A B) (B A) 5

6 More Basic Set Operations 1. If A B, what is A B? 2. If A B, what is A B? 3. When is A B =? 4. Do sets A B exist such that A B = B A? 5. Are there sets A and B such that A B = {(1, 2)}? 6. Are there sets A and B such that A B = {(1, 2), (1, 5), (2, 3), (2, 5), (3, 3), (3, 5)}? Inclusion-Exclusion two sets, basic In a group of 20 students, 11 play the piano and 7 play the guitar. If two play both the piano and the guitar, how many play: 1. at least one of these instruments (piano or guitar)? 2. neither instrument (neither the piano nor the guitar)? Venn Diagrams True or false: The oldest chess player among mathematicians and the oldest mathematician amongst chess players can be two different people. The best chess player among mathematicians and the best mathematician among chess players can be two different people. 6

7 Equality of Sets State True or False for each of the following. Justify your argument by either giving a proof or a counter-example. You may use Venn diagrams to present your argument. 1. (A B) C = A (B C) 2. (A B) = A B 3. (A C) B = A (C B) 4. (A C) (A B) = A (B C) 5. (A (B C)) ((A C) (D C) (B C)) = ((A B) (C A) (A B)) (((A D) B) C) 7

8 EXTRA: Cutting parts from a large sheet Can one cut three 3x3 squares and six 2x3 rectangles from a sheet of paper 8x8 square? Can one cut four 3x3 squares and eight 2x4 rectangles from a sheet of paper 10x10 square? Can one cut four 4x4 squares and and six 2x4 rectangles from a sheet of paper 11x11 square? EXTRA: remainders mod 15 Given any 7 integers, show that there are 2 of them with either sum or difference or product = multiple of 15. 8