CSE 20 DISCRETE MATH Prof. Shachar Lovett Clicker frequency: CA

Todays topics Set operations  Vs  Venn diagrams Sets equality and how to prove it Power set, Cartesian product

Set operations JS p. 47

 Vs  Which one of the following is true? A. 1  {1,2,3} B. 1  {1,2,3} C. {1}  {1,2,3} D. {1}  1,2,3 E. None/other/more than one

 Vs  Recall: x  A: x is an element in the set A A  B: A is a subset of B (all elements of A are also elements of B) Examples: 1  {1,2,3} {5,7}  {5,6,7} Elements can also be set! {1,3}  {{2}, 4, {1,3}} We can have set of sets of sets of sets …

 Vs  Which one of the following is true? A. 1  {{1},{2},{3}} B. 1  {{1},{2},{3}} C. {1}  {{1},{2},{3}} D. {1}  {{1},{2},{3}} E. None/other/more than one

 Vs  Which one of the following is true? A.   { ,{  },{{  }}} B.   { ,{  },{{  }}} C. {  }  { ,{  },{{  }}} D. {  }  { ,{  },{{  }}} E. None/other/more than one

Venn diagrams An useful way to understand sets intersection & union Generic Venn diagram for 3 sets: Describes all possible 8=2 3 combinations for whether an element is in A or not; in B or not; in C or not U B A C

Venn diagrams U B A C

U B A C

U B A C

U B A C

Set equality

Proving set equality

Proving set equality: simple example X = {n  N: n is even} Y = {n  N: n+1 is odd} Claim: X=Y Proof that X  Y: If n  X, then n is even, hence n+1 is odd, hence n+1  Y Proof that Y  X: If n  Y, then n+1 is odd, hence n is odd, hence n  X

Proving set equality: another example

Power Set JS p. 45

Power Set Which one of the following is always true? A. A  P(A) B. A  P(A) C. {A}  P(A) D. {A}  P(A) E. None/other/more than one

Cartesian product JS p. 48

Cartesian product

Next class More about sets Read section 2.1 in Jenkyns, Stephenson