# 2. Relations and Functions Class 11 Maths

“The first step toward success is taken when you refuse to be a captive of the environment in which you first find yourself.” â€“ Mark Caine

## RELATIONS AND FUNCTIONS CLASS 11 MATHS

In this Chapter, we studied about relations and functions.The main features of this Chapter are as follows:

Ordered pair A pair of elements grouped together in a particular order.

Cartesian product A Ã— B of two sets A and B is given by A Ã— B = {(a, b): a âˆˆ A, b âˆˆ B} In particular R Ã— R = {(x, y): x, y âˆˆ R} and R Ã— R Ã— R = (x, y, z): x, y, z âˆˆ R}

If (a, b) = (x, y), then a = x and b = y.

If n(A) = p and n(B) = q, then n(A Ã— B) = pq.

A Ã— Ï† = Ï†

In general, A Ã— B â‰  B Ã— A.

Relation A relation R from a set A to a set B is a subset of the cartesian product A Ã— B obtained by describing a relationship between the first element x and the second element y of the ordered pairs in A Ã— B.

The image of an element x under a relation R is given by y, where (x, y) âˆˆ R,

The domain of R is the set of all first elements of the ordered pairs in a relation R.

The range of the relation R is the set of all second elements of the ordered pairs in a relation R.

Function A function f from a set A to a set B is a specific type of relation for which every element x of set A has one and only one image y in set B. We write f: Aâ†’B, where f(x) = y.

A is the domain and B is the codomain of f.

The range of the function is the set of images.

A real function has the set of real numbers or one of its subsets both as its domain and as its range.

Algebra of functions For functions f : X â†’ R and g : X â†’ R, we have
(f + g) (x) = f (x) + g(x), x âˆˆ X
(f â€“ g) (x) = f (x) â€“ g(x), x âˆˆ X
(f.g) (x) = f (x) .g (x), x âˆˆ X
(kf) (x) = k ( f (x) ), x âˆˆ X, where k is a real number.
(f/g) (x) =f(x)/g(x), x âˆˆ X, g(x) â‰  0