Sets

Set membership
Set builder notation
Cardinal number
Ordinal number
Equal sets
Subsets
The empty set or the null set
Universal set or universe
Power set

Definition of a Set and Notation
Set is a finite or infinite collection of distinct objects. Those objects are called elements or members of the set.
The notation,     A = { a, b, c means A equals the set of elements a, b and c.
Set membership
If B is a set and d is an object, then the relationship
d is an element of B we denote as d Î B, or
d not Î of B if d is not an element of B, i.e., d does not belong to the set B.
Set builder notation
A set can be described by specifying the properties which determines its elements, for example
S = { n Î N  |  3 n 8
denotes the set S of all natural numbers n such that n is in the range from 3 to 8 inclusive that is,
S = { 3, 4, 5, 6, 7, 8 }.
Cardinal number
A measure of the size of a set that does not take into account the order of its members, i.e., which only specifies the total number of its elements is called the cardinal number.
Ordinal number
A measure of a set that takes account of the order as well as the number of its elements is called the ordinal number.
Equal sets
Two sets A and B are said to be equal if they have the same elements. This is written A = B.
Subsets
If set S and T are two sets such that every element of S is also an element of the set T, then S is said to be the subset of the set T, and is written as S  Í T. Every set is a subset of itself.
If S is a subset of T, but not equal to T, then S is also proper subset of T. Therefore, a proper subset is the one strictly contained within a larger set, excluding some elements of the larger set.
The empty set or the null set
A set which has no members is called the empty set or the null set and is denoted by ø or { }.
The empty set is therefore a subset of every set.
Universal set or universe
A universal set U is a set large enough to include all the elements of any set relevant to a problem under consideration. Hence, the universal set is a set defined by the context.
Power set
A power set is a set of which the elements are all the subsets of a given set S including the empty set, written P (S) or 2S.
 Example: If S is the set { 1, 2, 3 } then power set of S,  P (S) = { {}, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}}.