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Relations and Functions
Question
Wired Faculty App

 Check the injectivity and surjectivity of the following functions :

(i) f : N → N given by f (x) = x2
(ii)    f : Z → Z given by f (x) = x2
(iii)    f : R → R given by f (x) = x2 (iv) f : N → N given by f (x) = x3
(v) f : Z → Z given by f (x) = x3

Answer

(i) f : N → N is given by f (x) = x2
Let x1 x2 ∈ N be such that f(x1) = f(x2)
∴ x12 = x 22 ⇒ x2 2 x1 2 = 0
⇒ (x2 –x1) (x2 + x1) = 0
⇒    x2 – x1 = 0    [x1 + x2 ≠ 0 as x1, x2 ∈ N]
⇒    x2 = x1 ⇒ x1 = x2
∴ f is one-one, i.e.. f is injective.
Since range of f = { 12 , 22, 32............}
= {1.4.9......} ≠ N.
∴ f is not surjective.
(ii) f : Z → Z is given by f (x) = x2
Let x1, x2 ∈ Z be such that f(x1)= f (x2)
∴ x22 =x22 ⇒ x22 = 0
⇒ (x2 – x1) (x2 + x1) = 0
⇒    x2 = x1    or x2 = – x1
∴ f (x1) = f(–x1) ∀ x1 ∈ Z
∴ f is not one-one, i.e. f is not injective.
Also range of f = { 02, 12, 22,.....}
= {0, 1,4, 9,.........}
≠     Z
∴ f is not onto i.e.. f is not surjective (iii) f : R → R is given by f (x1) = x2 Let x1, x2 ∈ R be such that f (x1) = f (x2)
⇒    x12 = x22 ⇒ (x2 – x2) (x2 + x1) = 0
⇒    x2 = x1 or x2 = – x1
⇒    f(x1) = f (–x1) ∀ x1 ∈ R
∴ f is not one-one, i.e., f is not injective.
As range of f does not contain any negative real, therefore, range of ≠ R.
Hence. f is not onto, i.e., f is not surjective.
(iv) f : N → N is given by f (x) = x3 Let x1 ,. x2 ∈ N be such that f (x1) = f(x2)
⇒    x13 = x23 ⇒ x1 = x2
∴ f is one-one, i.e., injective.
Also range of f = {13, 23, 33,.........}
= {1,8,27,.....}
≠ N
∴ f is not onto, i.e.,f is not surjective.
(v) f : Z → Z is given by f (x) = x3 Let x1, x2 ∈ Z be such that f (x1) = f (x2)
⇒    x13 = x23 ⇒ x1 = x2
Also range of f = {03 ± 13, ± 23, ± 33,....}
= {0, ± 1, ± 8, ± 27.............}
≠ Z ∴ f is not onto, i.e., f is not surfective.

Sponsor Area

Some More Questions From Relations and Functions Chapter

If a matrix has 24 elements, what are the possible orders it can have ? Wh'at. if it has 13 elements ?

lf a matrix has 18 elements, what are the possible orders it can have ? What, if it has 5 elements?

If a matrix A has 12 elements, what arc the possible orders it can have 7 What if it has 7 elements ?

Let A be the set of all students of a boys school. Show that the relation R in A given by R = {(a, b) : a is sister of b} is the empty relation and R’ = {(a, b) : the difference between heights of a and b is less than 3 meters} is the universal relation.

Show that the relation R in the set {1, 2, 3} given by R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3)} is reflexive but neither symmetric nor transitive.

Show that the relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} is symmetric but neither reflexive nor transitive.

Give an example of a relation which is

(i) Symmetric but neither reflexive nor transitive.
(ii) Transitive but neither reflexive nor symmetric.
(iii) Reflexive and symmetric but not transitive.
(iv) Reflexive and transitive but not symmetric.
(v) Symmetric and transitive but not reflexive.

Let L be the set of all lines in a plane and R be the relation in L defined as R = {(L1, L2) : L1 is perpendicular to L2}. Show that R is symmetric but neither reflexive nor transitive.

 Determine whether each of the following relations are reflexive, symmetric and transitive :

(i) Relation R in the set A = {1, 2, 3,....., 13, 14} defined as

R = {(x, y) : 3 x – y = 0}

(ii) Relation R in the set N of natural numbers defined as R = {(x, y) : y = x + 5 and x < 4} (iii) Relation R in the set A = {1, 2, 3, 4, 5, 6} as R = {(x,y) : y is divisible by x} (iv) Relation R in the set Z of all integers defined as R = {(x,y) : x – y is an integer}

(v) Relation R in the set A of human beings in a town at a particular time given by
(a)    R = {(x, y) : x and y work at the same place}
(b)    R = {(x,y) : x and y live in the same locality}
(c)    R = {(x, y) : x is exactly 7 cm taller than y}
(d)    R = {(x, y) : x is wife of y}
(e)    R = {(x,y) : x is father of y}