1. Week 7 - Monday
CS322

2. Last time
What did we talk about last time?
Sets

3. Questions?

4. Logical warmup A man offers you a bet He shows you three cards
One is red on both sides
One is green on both sides
One is red on one side and green on the other
He will put one of the cards, at random, on the table
If you can guess the color on the other side, you win
If you bet $100
You gain $25 on a win
You lose your $100 on a loss
Should you take the bet? Why or why not?

5. Set Theory

6. Set operations
We usually discuss sets within some superset U called the universe of discourse
Assume that A and B are subsets of U
The union of A and B, written A  B is the set of all elements of U that are in either A or B
The intersection of A and B, written A  B is the set of all elements of U that are in A and B
The difference of B minus A, written B – A, is the set of all elements of U that are in B and not in A
The complement of A, written Ac is the set of all elements of U that are not in A

7. The empty set
There is a set with no elements in it called the empty set
We can write the empty set { } or 
It comes up very often
For example, {1, 3, 5}  {2, 4, 6} = 
The empty set is a subset of every other set (including the empty set)

8. Disjoint sets and partitions
Two sets A and B are considered disjoint if A  B = 
Sets A1, A2, … An are mutually disjoint (or nonoverlapping) if Ai  Aj =  for all i  j
A collection of nonempty sets {A1, A2, … An} is a partition of set A iff:
A = A1  A2  …  An
A1, A2, … An are mutually disjoint

9. Power set
Given a set A, the power set of A, written P(A) or 2A is the set of all subsets of A
Example: B = {1, 3, 6}
P(B) = {, {1}, {3}, {6}, {1,3}, {1,6}, {3,6}, {1,3,6}}
Let n be the number of elements in A, called the cardinality of A
Then, the cardinality of P(A) is 2n

10. Cartesian product
An ordered n-tuple (x1, x2, … xn) is an ordered sequence of n elements, not necessarily from the same set
The Cartesian product of sets A and B, written A x B is the set of all ordered 2-tuples of the form (a, b), a  A, b  B
Thus, (x, y) points are elements of the Cartesian product R x R (sometimes written R2)

11. Subset Relations

12. Basic subset relations
Inclusion of Intersection:
For all sets A and B
A  B  A
A  B  B
Inclusion in Union:
A  A  B
B  A  B
Transitive Property of Subsets:
If A  B and B  C, then A  C

13. Element argument The basic way to prove that X is a subset of Y
Suppose that x is a particular but arbitrarily chosen element of X
Show that x is an element of Y
If every element in X must be in Y, by definition, X is a subset of Y

14. Procedural versions
We want to leverage the techniques we've already used in logic and proofs
The following definitions help with this goal:
x  X  Y  x  X  x  Y
x  X  Y  x  X  x  Y
x  X – Y  x  X  x  Y
x  Xc  x  X
(x, y)  X  Y  x  X  y  Y

15. Example proof Theorem: For all sets A and B, A  B  A Proof:
Let x be some element in A  B
x  A  x  B
x  A
Thus, all elements in A  B are in A
A  B  A
QED
Premise
Definition of intersection
Specialization
By generalization
Definition of subset

16. Laying down the law (again)
Name
Law
Dual
Commutative
A  B = B  A
A  B = B  A
Associative
(A  B)  C = A  (B  C)
(A  B)  C = A  (B  C)
Distributive
A  (B  C) = (A  B)  (A  C)
A  (B  C) = (A  B)  (A  C)
Identity
A   = A
A  U = A
Complement
A  Ac = U
A  Ac = 
Double Complement
(Ac)c = A
Idempotent
A  A = A
A  A = A
Universal Bound
A  U = U
A   = 
De Morgan’s
(A  B)c = Ac  Bc
(A  B)c = Ac  Bc
Absorption
A  (A  B) = A
A  (A  B) = A
Complements of U and 
Uc = 
c = U
Set Difference
A – B = A  Bc

17. Proving set equivalence
To prove that X = Y
Prove that X  Y and
Prove that Y  X

18. Example proof of equivalence
Theorem: For all sets A,B, and C, A  (B  C) = (A  B)  (A  C)
Proof:
Let x be some element in A  (B  C)
x  A  x  (B  C)
Case 1: Let x  A
x  A  x  B
x  A  B
x  A  x  C
x  A  C
x  A  B  x  A  C
x  (A  B)  (A  C)
Case 2: Let x  B  C
x  B  x  C
x  B
x  A  x  B
x  A  B
x  C
x  A  x  C
x  A  C
x  A  B  x  A  C
x  (A  B)  (A  C)
In all possible cases, x  (A  B)  (A  C), thus A  (B  C)  (A  B)  (A  C)

19. Proof of equivalence continued
Let x be some element in (A  B)  (A  C)
x  (A  B)  x  (A  C)
Case 1: Let x  A
x  A  x  B  C
x  A  (B  C)
Case 2: Let x  A
x  A  B
x  A  x  B
x  B
x  A  C
x  A  x  C
x  C
x  B  x  C
x  B  C
x  A  x  B  C
x  A  (B  C)
In all possible cases, x  A  (B  C), thus (A  B)  (A  C)  A  (B  C)
Since both A  (B  C)  (A  B)  (A  C) and (A  B)  (A  C)  A  (B  C), A  (B  C) = (A  B)  (A  C)
QED

20. Proof example Prove that, for any set A, A   = 
Hint: Use a proof by contradiction

21. Disproofs and Algebraic Proofs

22. Disproving a set property
Like any disproof for a universal statement, you must find a counterexample to disprove a set property
For set properties, the counterexample must be a specific examples of sets for each set in the claim

23. Counterexample example
Claim: For all sets A, B, and C, (A – B)  (B – C) = A – C
Find a counterexample

24. Algebraic set identities
We can use the laws of set identities given before to prove a statement of set theory
Be extremely careful (even more careful than with propositional logic) to use the law exactly as stated

25. Algebraic set identity example
Theorem: A – (A  B) = A – B
Proof:
A – (A  B) = A  (A  B) c
= A  (Ac  B c)
= (A  Ac)  (A  B c)
=   (A  B c)
= A  B c
= A – B
QED

26. Prove or disprove
For all sets A, B, and C, if A  B and B  C, then A  C

27. Prove or disprove
For all sets A and B, ((Ac  Bc) – A)c = A

28. Upcoming

29. Next time… Russell's paradox Halting program Functions
Composition of functions
Cardinality

30. Reminders
Homework 5 is due Friday
Read Chapter 7