Set Theory

1
Set Theory
2
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,

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.

A
B
A
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,
• 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.

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 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

6
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.

7
Set Properties

• Commutative Laws
• Associative Laws
• Distributive Laws

8
Set Properties

• Double Complement Law
• De Morgans Laws
• Absorption Laws

9
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 ?

10
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.

11
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.

12
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.