Elementary Set Theory

Elementary set is a collection of arbitrary objects and is generally represented by the capital letters A, B, etc. For e.g.:

A = {chair, table, man, woman}

B = (1, 2, 3, 4, 5, 6, 7, 8)

C = {the set of all the insects in the world}

are some of the examples of sets. The objects that are contained in a set are known as its elements or members. In some object, X is a member of a set, say A, it is written as x? A and is referred to as x belongs to A or x is in A.  x? A means x is not a member of A.

A set may contain a limited or an unlimited number of elements. A set which contains a limited number of elements is known as 'finite set' and the one which contains an unlimited number of element is known as an 'infinite set'. The above set A and B are finite sets while the set C is an infinite set.

If a set does not contain any element, it is known as a 'null set' or an 'empty set' and it is represented by a special symbol Φ (phi).

For mathematical work we normally use the sets of numbers which are as follows:

(i) The set of natural numbers:

N = {1, 2, 3, 4, 5......}

(ii) The set of whole numbers:

W = {0, 1, 2, 3, 4.............}

(iii) The set of integers:

 I = {.......................- 3, -2, -1, 0, 1, 2, 3 ......................}

(iv) The set of rational numbers:

 Q = {a/a/ a b? I, b ≠ 0}

Sub-set

If each element of set A is an element of set B, the set A is known as the sub-set of B and is written as A B. Every set is a sub-set of itself & ? is a sub-set of every set.

For example: A = {1, 2, 3, 4,}, B = {1, 2, 3, 4.................}

Equal sets

(i. e. A = B).

Universal set

Every set is assumed to be a subset of a very big set, which is known as the universal set for the given set. For example, the set N = (1, 2, 3, 4...........} can be taken as the universal set for every set of counting numbers. A universal set is generally represented by U or S.