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For example ,

A = { All books of Algebra in your library }

B = { All books of Geometry in your library }

C = { All books in your library }

Here, the set C is taken as the universal set .

Complement of a set : Let È be the universal set and set A Ì B. The complement of the set A with respect to the universal set È is denoted by A˘ is defined as

A˘ or = { x | x ÎÈ but x Ï A }

For example

È = { 1, 2, 3, 4, . . . } and A = { 1, 3, 5, 7, . . . }
then A˘ = { 2, 4, 6, 8, . . . }

Equivalent sets :
Two sets are equivalent if and only if, a 1 - 1 correspondence exists between them.

For example

1. If A = { 1, 2, 3 }, B = {a, b, c } then A and B are 1-1. equivalent, since the correspondence between them is
2. If A = { x | x ÎN, x < 5} B = { x | x is a set of the word Dear }, then A and B are equivalent.

Note : If two sets are equal, then they must be equivalent. But two equivalent sets need not be equal.

Power set : The set of all subsets of a set A is the power set of the set A. It is denoted by p(A)

For example A = {1, 2}.
The power set of A is p(A) = {{1}, {2}, {1,2}, { }}

Note : The number of elements are 2 and number of subsets are 4. Then 4 = 22 Þ if there are 'n' number of elements in any set, the number of its subsets is given by 2n .

Cardinal Number of a set : The cardinal number of a finite set 'A' is the number of elements of the set A. It is denoted by n (A).

For example, If A = { 1, 2, 3 } , B = {a, b, c} then n (A) = 3 and n (B) = 3.

Thus n (A) = n (B) but n (A) = n (B) does not imply that A = B. Also n (f ) = 0.

Difference between two set : The set of all elements belonging to a set A but not belonging to a set B. It is written as A - B.

Index

2.1 Sets
2.2 Operations on Sets
2.3 The Algebra of Sets

Chapter 3