1. Set Theory

2. Notation S={a, b, c} refers to the set whose elements are a, b and c.
aS means “a is an element of set S”.
dS means “d is not an element of set S”.
{x S | P(x)} is the set of all those x from S such that P(x) is true. E.g., T={x Z | 0<x<10} .
Notes:
1) {a,b,c}, {b,a,c}, {c,b,a,b,b,c} all represent the same set.
2) Sets can themselves be elements of other sets, e.g., S={ {Mary, John}, {Tim, Ann}, …}

3. Relations between sets
Definition: Suppose A and B are sets. Then
A is called a subset of B: A  B
iff every element of A is also an element of B.
Symbolically,
A B  x, if xA then x B.
A  B  x such that xA and xB. B A A A B B
A  B
A  B
A  B

4. Relations between sets
Definition: Suppose A and B are sets. Then
A equals B: A = B
iff every element of A is in B and
every element of B is in A.
Symbolically,
A=B  AB and BA .
Example: Let A = {mZ | m=2k+3 for some integer k};
B = the set of all odd integers.
Then A=B.

5. Operations on Sets Definition: Let A and B be subsets of a set U.
1. Union of A and B: A  B = {xU | xA or xB}
2. Intersection of A and B:
A  B = {xU | xA and xB}
3. Difference of B minus A: BA = {xU | xB and xA}
4. Complement of A: Ac = {xU | xA}
Ex.: Let U=R, A={x R | 3<x<5}, B ={x R| 4<x<9}. Then
1) A  B = {x R | 3<x<9}.
2) A  B = {x R | 4<x<5}.
3) BA = {x R | 5 ≤x<9}, AB = {x R | 3<x ≤4}.
4) Ac = {xR | x ≤3 or x≥5}, Bc = {xR | x ≤4 or x≥9}

6. Properties of Sets Theorem 1 (Some subset relations): 1) AB  A
2) A  AB
3) If A  B and B  C, then A  C .
To prove that A  B use the “element argument”:
1. suppose that x is a particular but arbitrarily chosen element of A,
2. show that x is an element of B.

7. Proving a Set Property Theorem 2 (Distributive Law):
For any sets A,B and C:
A  (B  C) = (A  B)  (A  C) .
Proof: We need to show that
(I) A  (B  C)  (A  B)  (A  C) and
(II) (A  B)  (A  C)  A  (B  C) .
Let’s show (I).
Suppose x  A  (B  C) (1)
We want to show that x  (A  B)  (A  C) (2)

8. Proving a Set Property Proof (cont.):
x  A  (B  C)  x  A or x  B  C .
(a) Let x  A. Then
x  AB and x  AC  x  (A  B)  (A  C)
(b) Let x  B  C. Then xB and xC.
Thus, (2) is true, and we have shown (I).
(II) is shown similarly (left as exercise). ■

9. Set Properties
Commutative Laws:
Associative Laws:
Distributive Laws:

10. Set Properties Double Complement Law: De Morgan’s Laws:
Absorption Laws:

11. Showing that an alleged set property is false
Statement: For all sets A,B and C,
A(BC) = (AB)C .
The following counterexample
shows that the statement is false.
Counterexample:
Let A={1,2,3,4} , B={3,4,5,6} , C={3}.
Then BC = {4,5,6} and A(BC) = {1,2,3} .
On the other hand,
AB = {1,2} and (AB)C = {1,2} .
Thus, for this example
A(BC) ≠ (AB)C .

12. Empty Set The unique set with no elements
is called empty set and denoted by .
Set Properties that involve  .
For all sets A,
1.   A
2. A   = A
3. A   = 
4. A  Ac = 

13. Disjoint Sets A and B are called disjoint iff A  B =  .
Sets A1, A2, …, An are called mutually disjoint
iff for all i,j = 1,2,…, n
Ai  Aj =  whenever i ≠ j .
Examples:
1) A={1,2} and B={3,4} are disjoint.
2) The sets of even and odd integers are disjoint.
3) A={1,4}, B={2,5}, C={3} are mutually disjoint.
4) AB, BA and AB are mutually disjoint.

14. Partitions Definition: A collection of nonempty sets
{A1, A2, …, An} is a partition of a set A iff
1. A = A1  A2  …  An
2. A1, A2, …, An are mutually disjoint.
Examples:
1) {Z+, Z-, {0} } is a partition of Z.
2) Let S0={n  Z | n=3k for some integer k}
S1={n  Z | n=3k+1 for some integer k}
S2={n  Z | n=3k+2 for some integer k}
Then {S0, S1, S2} is a partition of Z.

15. Power Sets Definition: Given a set A,
the power set of A, denoted P (A) ,
is the set of all subsets of A.
Example: P ({a,b}) = {, {a}, {b}, {a,b}} .
Properties:
1) If A  B then P (A)  P (B) .
2) If a set A has n elements
then P (A) has 2n elements.