CSE 20 DISCRETE MATH Prof Shachar Lovett http

CSE 20 DISCRETE MATH Prof. Shachar Lovett http: //cseweb. ucsd. edu/classes/wi 15/cse 20 -a/ Clicker frequency: CA

Todays topics • Set sizes • Set builder notation • Set rapid-fire quiz • Section 2. 1 in Jenkyns, Stephenson

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

Union size •

Intersection size •

Cartesian product size •

Important sets of numbers • Z = integers Z = {…, -3, -2, -1, 0, 1, 2, 3, …} • N = natural numbers = positive integers N = {1, 2, 3, …} • Q = rational numbers Q = {x/y : x, y Z}

Set builder notation •

Set builder notation •

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 | <some criteria>} • + can be used for large or infinite sets, clearly sets forth rules for membership

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

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

Russell’s paradox •

Set Theory rapid-fire practice •

Set Theory rapid-fire practice •

Set Theory rapid-fire practice •

Next class • Functions, sequences • Read section 2. 2 in Jenkyns, Stephenson