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# 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.” – Mark Caine

## RELATIONS AND FUNCTIONS CLASS 11 MATHS

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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 × B of two sets A and B is given by A × B = {(a, b): a ∈ A, b ∈ B} In particular R × R = {(x, y): x, y ∈ R} and R × R × R = (x, y, z): x, y, z ∈ R}

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

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

A × φ = φ

In general, A × B ≠ B × A.

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

The image of an element x under a relation R is given by y, where (x, y) ∈ 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: A→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.

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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 → R and g : X → R, we have

(f + g) (x) = f (x) + g(x), x ∈ X

(f – g) (x) = f (x) – g(x), x ∈ X

(f.g) (x) = f (x) .g (x), x ∈ X

(kf) (x) = k ( f (x) ), x ∈ X, where k is a real number.

(f/g) (x) =f(x)/g(x), x ∈ X, g(x) ≠ 0

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