# NCERT Class 11-Math's: Chapter –2 Relations and Functions Part 1

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2.1 Overview

This chapter deals with linking pair of elements from two sets and then introduce relations between the two elements in the pair. Practically in every day of our lives, we pair the members of two sets of numbers. For example, each hour of the day is paired with the local temperature reading by T.V. Station’s weatherman, a teacher often pairs each set of score with the number of students receiving that score to see more clearly how well the class has understood the lesson. Finally, we shall learn about special relations called functions.

2.1.1 Cartesian products of sets

Definition: Given two non-empty sets A and B, the set of all ordered pairs , where and is called Cartesian product of A and B; symbolically, we write

If and , then

and

(i) Two ordered pairs are equal, if and only if the corresponding first elements are equal and the second elements are also equal, i.e. if and only if .

(ii) If .

(ii) . Here is called an ordered triplet.

2.1.2 Relations

A Relation R from a non-empty set A to a non-empty set B is a subset of the Cartesian product set A × B. The subset is derived by describing a relationship between the first element and the second element of the ordered pairs in .

The set of all first elements in a relation R, is called the domain of the relation R, and the set of all second elements called images, is called the range of R.

For example, the set is a relation; the domain ofand the range of .

(i) A relation may be represented either by the Roster form or by the set builder form, or by an arrow diagram which is a visual representation of a relation.

(ii) If ; then the and the total number of possible relations from the set A to set .

2.1.3 Functions

A relation f from a set A to a set B is said to be function if every element of set A has one and only one image in set B.

In other words, a function f is a relation such that no two pairs in the relation has the same first element.

The notation means that f is a function from X to is called the domain of f and Y is called the co-domain of . Given an element x ∈ X, there is a unique element y in Y that is related to x. The unique element y to which f relates x is denoted by f (x) and is called f of x, or the value of f at x, or the image of x under f.

The set of all values of f (x) taken together is called the range of f or image of X under f. Symbolically.

range of

Definition : A function which has either R or one of its subsets as its range, is called a real valued function. Further, if its domain is also either R or a subset of R, it is called a real function.

2.1.4 Some specific types of functions

(i) Identity function:

The function f : R → R defined by y = f (x) = x for each x ∈ R is called the identity function.

(ii) Constant function: The function defined by , where C is a constant ∈ R, is a constant function.

(iii) Polynomial function: A real valued function defined by , where , and , for each, is called Polynomial functions.

(iv) Rational function: These are the real functions of the type, where and are polynomial functions of x defined in a domain, where . For example defined by is a rational function.

(v) The Modulus function: The real function defined by

is called the modulus function.

Domain of

Range of

(vi) Signum function: The real function

defined by

is called the signum function. Domain of , Range of

(vii) Greatest integer function: The real function defined by assumes the value of the greatest integer less than or equal to x, is called the greatest integer function.

Thus