CSE 20 DISCRETE MATH Prof. Shachar Lovett 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)| = 2n 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

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 { …} 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

Primes Enumeration may not be clear: { …} 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

Next class Functions, sequences Read section 2.2 in Jenkyns, Stephenson