1. CSE 20: Discrete Mathematics for Computer Science Prof. Shachar Lovett

2. Today’s Topics: 1. Set sizes 2. Set builder notation 3. Rapid-fire set-theory practice 2

3. 1. Set sizes 3

4. Power set  Let A be a set of n elements (|A|=n)  How large is P(A) (the power-set of A)? A. n B. 2n C. n 2 D. 2 n E. None/other/more than one 4

5. Cartesian product  |A|=n, |B|=m  How large is A x B ? A. n+m B. nm C. n 2 D. m 2 E. None/other/more than one 5

6. Union  |A|=n, |B|=m  How large is A  B ? A. n+m B. nm C. n 2 D. m 2 E. None/other/more than one 6

7. Intersection  |A|=n, |B|=m  How large is A  B ? A. n+m B. nm C. At most n D. At most m E. None/other/more than one 7

8. 2. Set builder notation 8

9. Set builder notation 9

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11. Ways of defining a set  Enumeration:  {1,2,3,4,5,6,7,8,9}  + very clear  - impractical for large sets  Incomplete enumeration (ellipses):  {1,2,3,…,98,99,100}  + takes up less space, can work for large or infinite sets  - not always clear  {2 3 5 7 11 13 …} What does this mean? What is the next element?  Set builder:  { n | }  + can be used for large or infinite sets, clearly sets forth rules for membership 11

12. Primes  Enumeration may not be clear:  {2 3 5 7 11 13 …}  How can we write the set Primes using set builder notation? A. {n  N :   a,b  N, n=ab} B. {n  N :  a,b  N, n=ab  (a=1  b=1)} C. {a,b  N :  n  N, n=ab  (a=n  b=n)} D. {n  N :  a,b  N, n=ab  (a=1  b=1)} E. None/other/more than one 12

13. Russell’s paradox  Let A={S| S  S}  Does A  A? A. Yes B. No C. Neither D. Both E. Other 13

14. Russell’s paradox 14

15. 3. Rapid-fire set-theory practice Clickers ready! 15

16. Set Theory rapid-fire practice 16

17. Set Theory rapid-fire practice 17

18. Set Theory rapid-fire practice 18