Introduction to Set

Set

A well-defined collection of objects is called at set.

The types of Representation are:

(i) Roaster form

(ii) Set builder form

Types of Set

1.  Subset – Let A and B be two sets such that every element of A is in B, then we say that A is a subset of B and we write, A Í B.

2.  Equal sets – Two sets A and B are said to be equal, if every element of A is in B and every element of B is in A, and we write, A = B.

Note 1 – Repetition of elements, in a set does not affect the equality of sets e.g. {1, 2, 3, 1, 2, 3,…) = {1, 2, 3}

Note 2 – The order in which the elements of a set are listed is immaterial e.g.

3.  Proper subset – If A Í B and A ¹ B, then A is called a proper subset of B and we write, A Ì B.

4.  Power set – The set of all subsets of a given set A is called the power set of A, to be denoted by P (A).

5.  Singleton Set– A set containing one element is called Singleton Set.

6.  Universal Set – The set which contains all the elements under consideration in a particular problem is called the universal set denoted by S. Suppose that P is a subset of S. Then the complement of P, written as Pc (or P’) contains all the elements in S but not in P. This can also be written as S – P or S ~ P.

7.  Equivalent Set – Two finite sets A & B are said to be equivalent if n (A) = n (B).

Note: Equal sets are equivalent but equivalent sets need not be equal.

8.  If A contains n elements, then the number of proper subsets of A is (2n –1).

9.  If A contains n elements, then the number of subset of A = 2 n.

10.       Cardinal number – The number of distinct elements contained in a finite set A is called its cardinal number. It is denoted by n (A).

Operations on sets

[Symmetric difference]

De-Morgan’s Laws

Important results:

Cartesian product of sets

Results on Cartesian product of sets:

(iv) If A and B have n elements in common, then A ´ B and B ´ A have n2 elements in common.