The concept of set serves as a fundamental part of the present day mathematics. Today this concept is being used in almost every branch of mathematics. Sets are used to define the concepts of relations and functions. The study of geometry, sequences, probability requires the knowledge of sets.
The theory of sets was developed by German mathematician Georg Cantor (1845-1918). He first encountered sets while working on "problems on trigonometric series".
A set is a well-defined collection of objects.
A set which does not contain any element is called empty set. A set which consists of a definite number of elements is called finite set, otherwise, the set is called infinite set.
Two sets A and B are said to be equal if they have exactly the same elements. A set A is said to be subset of a set B, if every element of A is also an element of B. Intervals are subsets of R.
A power set of a set A is collection of all subsets of A. It is denoted by P(A).
Venn Diagrams: Most of the relationships between sets can be represented by means of diagrams which are known as Venn diagrams. Venn diagrams are named after the English logician, John Venn (1834-1883). These diagrams consist of rectangles and closed curves usually circles. The universal set is represented usually by a rectangle and its subsets by circles.
The union of two sets A and B is the set of all those elements which are either in A or in B. The intersection of two sets A and B is the set of all elements which are common. The difference of two sets A and B in this order is the set of elements which belong to A but not to B.
The complement of a subset A of universal set U is the set of all elements of U which are not the elements of A.
For any two sets A and B,(A ∪ B)′ = A′ ∩ B′ and (A ∩ B)′ = A′ ∪ B′
If A and B are finite sets such that A ∩ B = φ, then n(A ∪ B) = n(A) + n(B).
If A ∩ B ≠ φ, then n(A ∪ B) = n(A) + n(B) - n(A ∩ B)