Set Theory

Definition : A set having only one element is called a singleton.
Example: (i) A = {8}, (ii) {φ}
Theorem 1: Two sets A and B are equal if and only if A ⊆ B and B ⊆ A.
Proof: If A = B, every member of A is a member of B and every member of B is a member of A. Hence A ⊆ B and B ⊆ A
Conversely let us suppose that A ≠ B, then there is either an element of A that is not in B or there is an element of B that is not in A. But A ⊆ B , therefore every element of A is in B and B ⊆ A , therefore every element of B is in A. Therefore, our assumption that A ≠ B leads to a contradiction, hence A = B.
Theorem 2: If φ and φ ′ are empty sets, then φ = φ ′ .
Proof: Suppose φ ≠ φ ′ . Then one of the following statements must be true:
1. There is an element x ∈φ such that x ∉φ ′
2. There is an element x ∈φ ′ such that x ∉φ . But both these statements are false, since neither φ nor φ ′ has any elements. If follows that φ = φ

Finite Sets
Definition : A set is said to be finite, if it has finite number of elements.
(i) {1, 2, 3, 5}
(ii) The letters of the English alphabet.
Infinite Sets
Definition : A set is infinite, if it is not finite.
(i) The set of all real numbers.
(ii) The points on a line.
Universal Sets
Definition : In many discussions all the sets are considered to be subsets of one particular set. This set is called the universal set for that discussion. The Universal set is often designated by the script letter U (or by X). Universal set in not unique, and it may change from one discussion to another.
Example: If A = {0, 2, 7}, B = {3, 5, 6}, C = {1, 8, 9, 10} then the universal set can be taken as the set.
U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}.

The Power Set
Definition : The set of all subsets of a set A is called the power set of A.
The power set of A is denoted by P (A).
Hence P(A) = {x| x ⊆ A}

The power set of A is also denoted sometimes by 2A If A has n elements in it, then P (A) has 2n elements:
Example 1: If A = {a, b} then
P(A) = {φ , {a}, {b}, {a , b}}
Example 2: The empty set φ , has only subset, therefore P(φ ) = {φ}.
Note: A set is never equal to its power set. In the programming language Pascal, the notion power set is used to
define data type in the language.
Disjoint Set
Definition : Two sets are said to be disjoint if they have no element in common.

Example: The sets, A = {0, 4, 7, 9} and B = {3, 6, 10} are disjoint.