2.7 Summary

NCERT Class 11 Mathematics for blind and visually impaired students.

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 co-domain 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 → 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