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

2. Today’s Topics:  Vs  Venn Diagrams Proving two sets are equal
Power Set and Cross Product

3. Set notations Union: AB = {x:xA  xB}
Intersection: AB = {x:xA  xB}
Complement: Ac= {x:xA}
Difference: A-B = {x:xA  xB}
Empty set: 

4. 1.  Vs 
Don’t be confused!

5.  Vs  Which one of the following is true? 1  {1,2,3} 1  {1,2,3}
{1}  {1,2,3}
{1}  1,2,3
None/other/more than one

6.  Vs  Which one of the following is true? 1  {{1},{2},{3}}
1  {{1},{2},{3}}
{1}  {{1},{2},{3}}
{1}  {{1},{2},{3}}
None/other/more than one

7.  Vs  Which one of the following is true?   {,{},{{}}}
  {,{},{{}}}
{}  {,{},{{}}}
{}  {,{},{{}}}
None/other/more than one

8. 2. Venn Diagrams

9. If A, B, and C are subsets of U, then 𝐴−𝐶 ∩ 𝐵−𝐶 ∩ 𝐴−𝐵 =∅
We will prove this using Venn Diagram method
First, we need to draw a generic Venn Diagram: U B A C

10. If A, B, and C are subsets of U, then 𝐴−𝐶 ∩ 𝐵−𝐶 ∩ 𝐴−𝐵 =∅
Which area(s) represent 𝐵−𝐶 ?
1,2
2,5,8
1,2,3,4
3,4
Other/none/more U B A C

11. If A, B, and C are subsets of U, then 𝐴−𝐶 ∩ 𝐵−𝐶 ∩ 𝐴−𝐵 =∅
Which area(s) represent 𝐴−𝐶 ?
1,4
2,3
8,7
5,6
Other/none/more U B A C

12. If A, B, and C are subsets of U, then 𝐴−𝐶 ∩ 𝐵−𝐶 ∩ 𝐴−𝐵 =∅
Which area(s) represent 𝐴−𝐵 ?
A = 2,3,5,6
B = 3,4,6,7
A – B = 2,5
A-C = 2,3
B-C = 3,4
A-B = 2,5
There is no number that appears in all three lists, so the intersection is empty set U B A C

13. If A, B, and C are subsets of U, then 𝐴−𝐶 ∩ 𝐵−𝐶 ∩ 𝐴−𝐵 =∅
Algebraic proof:
𝐴−𝐶 ∩ 𝐵−𝐶 ∩ 𝐴−𝐵 =𝐴∩ 𝐶 𝐶 ∩𝐵∩ 𝐶 𝐶 ∩𝐴∩ 𝐵 𝐶 (by definition of set difference, and by associative law –removing parentheses)
=𝐴∩ 𝐶 𝐶 ∩ 𝐶 𝐶 ∩𝐴∩ (𝐵 𝐶 ∩𝐵) (by commutative law, and by associative law –adding parentheses)
=𝐴∩ 𝐶 𝐶 ∩ 𝐶 𝐶 ∩𝐴∩∅(by complement law)
=∅(by universal bound law)

14. 3. Proving two sets are equal

15. Set equality Given sets X and Y, X = Y means: ∀𝑧, 𝑧∈𝑋 ↔𝑧∈𝑌
∀𝑧, 𝑧∈𝑋 ↔𝑧∈𝑌
So we can use our proof template for “if and only if” to show that two sets are equal
(Your book shows a different way, “chain of iffs”)
Recall: 𝑝↔𝑞 is equivalent to 𝑝→𝑞 AND 𝑞→𝑝
“IFF” proof template says do this as two proofs:
Prove 𝑝→𝑞
Next, prove 𝑞→𝑝
(Each part uses the “proving implications” template)

16. Set equality: example X={even numbers} Y={n: n+2 is even} Claim: X=Y
Proof:
nX  n is even  n+2 is even  nY

17. Set equality
Another take: one way to prove that X=Y is to prove that both
X  Y
Y  X

18. Set equality: another example
X={n: 6 divides n}
Y={n:2 divides n and 3 divides n}
Claim: X=Y
XY: Let nX, i.e. n=6k. Hence, n=2(3k),n=3(2k), which means nY.
YX: Let nY, i.e. n=2a,n=3b. So, 3b=2a is even. Hence b must be even (need lemma: odd times odd is odd), i.e. b=2c. So, n=6c, hence nX.

19. Set equality: yet another example
Can also prove directly
Claim: (AC)C=A
Proof:
AC={x: x  A}
(AC)C = {x: x  AC} = {x: x  A}

20. 4. Power Set and Cross Product

21. Power Sets
For any set A, let P(A) be its power set, which is the set of all its subsets
Note that A is itself a member of P(A).
Recall: ∅ denotes the empty set.
Question: What is P({1,2})?
{1, 2, {1,2}}
{{1}, {2}, {1,2}, ∅}
{1, 2, {1,2}, {2,1}}
None/other/more than one

22. Cross Product
For any sets A and B, let A x B = {(x, y); x ∈ A and y ∈ B}
Question: What is {1,2} x {1, 2, 3} ?
{1, 2, 3}
{(1,1), (2,2), (∅ ,3)}
{(1,1),(1,2),(1,3),(2,1),(2,2),(2,3)}
{(1),(1,2),(1,3),(2),(2,3)}
None/other/more than one