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Set Theory

Notation

- Sa, 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., Tx ?Z

0ltxlt10 . - 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,

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.

A

B

A

B

A ? B

A ? B

A ? B

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,
- AB ? A?B and B?A .
- Example Let A m?Z m2k3 for some integer

k - B the set of all odd integers.
- Then AB.

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 UR, Ax ?R 3ltxlt5, B x ?R

4ltxlt9. Then - 1) A ? B x ?R 3ltxlt9.
- 2) A ? B x ?R 4ltxlt5.
- 3) B?A x ?R 5 xlt9, A?B x ?R 3ltx

4. - 4) Ac x?R x 3 or x5, Bc x?R x 4 or

x9

Set Properties

- 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.

Set Properties

- Commutative Laws
- Associative Laws
- Distributive Laws

Set Properties

- Double Complement Law
- De Morgans Laws
- Absorption Laws

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 ?

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) A1,2 and B3,4 are disjoint.
- 2) The sets of even and odd integers are

disjoint. - 3) A1,4, B2,5, C3 are mutually

disjoint. - 4) A?B, B?A and A?B are mutually disjoint.

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 S0n ? Z n3k for some integer k
- S1n ? Z n3k1 for some integer k
- S2n ? Z n3k2 for some integer k
- Then S0, S1, S2 is a partition of Z.

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.

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