Question 2

Check the injectivity and surjectivity of the following functions:

(i)* f* : **N → N** given by* f(x*) = x^{2}

(ii)* f* : **Z → Z** given by *f(x)* = x^{2}

(iii)* f* : **R → R** given by* f(x)* = x^{2}

(iv)* f *: **N → N** given by *f(x)* = x^{3}

(v)* f* : **Z → Z** given by *f(x)* = x^{3 }

Answer

(i) *f*: **N** → **N** is given by,

*f*(*x*) = *x*^{2}

It is seen that for *x*, *y* ∈**N**, *f*(*x*) = *f*(*y*) ⇒ *x*2 = *y*2 ⇒ *x* = *y*.

∴*f* is injective.

Now, 2 ∈ **N**. But, there does not exist any *x* in **N** such that *f*(*x*) = *x*^{2} = 2.

∴ *f* is not surjective.

Hence, function *f* is injective but not surjective.

(ii) *f*: **Z** → **Z** is given by,

*f*(*x*) = *x*^{2}

It is seen that *f*(-1) = *f*(1) = 1, but -1 ≠ 1.

∴ *f* is not injective.

Now,-2 ∈ **Z**. But, there does not exist any element *x* ∈**Z** such that *f*(*x*) = *x*^{2} = -2.

∴ *f* is not surjective.

Hence, function *f* is neither injective nor surjective.

(iii) *f*: **R** → **R** is given by,

*f*(*x*) = *x*^{2}

It is seen that *f*(-1) = *f*(1) = 1, but -1 ≠ 1.

∴ *f* is not injective.

Now,-2 ∈ **R**. But, there does not exist any element *x* ∈ **R** such that *f*(*x*) = *x*^{2} = -2.

∴ *f* is not surjective.

Hence, function *f* is neither injective nor surjective.

(iv) *f*: **N** → **N** given by,

*f*(*x*) = *x*^{3}

It is seen that for *x*, *y* ∈**N**, *f*(*x*) = *f*(*y*) ⇒ *x*^{3} = *y*^{3} ⇒ *x* = *y*.

∴*f* is injective.

Now, 2 ∈ **N**. But, there does not exist any element *x* in domain **N** such that *f*(*x*) = *x*^{3 }= 2.

∴ *f* is not surjective

Hence, function *f* is injective but not surjective.

(v) *f*: **Z** → **Z** is given by,

*f*(*x*) = *x*^{3}

It is seen that for *x*, *y* ∈ **Z**, *f*(*x*) = *f*(*y*) ⇒ *x*^{3} = *y*^{3} ⇒ *x* = *y*.

∴ *f* is injective.

Now, 2 ∈ **Z**. But, there does not exist any element *x* in domain **Z** such that *f*(*x*) = *x*^{3} = 2.

∴ *f* is not surjective.

Hence, function *f* is injective but not surjective.

- Q:- 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): 3x − 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 as 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} - Q:- Given 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. - Q:- Show that each of the relation R in the set A = { x ∈Z: 0≤x≤12}, A={x} given by

(i) R = { (a,b) : |a - b| is a multiple of 4}

(ii) R = {(a,b):a = b} is an equivalence relation.

Find the set of all elements related to 1 in each case. - Q:- Show that the relation R in the set R of real numbers, defined as R = {(a, b): a ≤ b
^{2}} is neither reflexive nor symmetric nor transitive. - Q:- Let L be the set of all lines in XY plane and R be the relation in L defined as R = {(L1, L2): L1 is parallel to L2}. Show that R is an equivalence relation. Find the set of all lines related to the line y = 2x + 4.
- Q:-
Prove that the Greatest Integer Function

*f*: R → R, given by f(x) = [x], is neither one-one nor onto, where [x] denotes the greatest integer less than or equal to x. - Q:- If a matrix has 24 elements, what are the possible order it can have? What, if it has 13 elements?
- Q:- Show that the relation R in R defined as R = {(a, b): a ≤ b}, is reflexive and transitive but not symmetric.
- Q:-
Show that the Modulus Function

*f*: R → R, given by f(x) = |x|, is neither oneone nor onto, where | x | is x, if x is positive or 0 and |x| is – x, if x is negative. - Q:- Show that the relation R in the set A = {1, 2, 3, 4, 5} given by R = { (a,b) ; |a - b| is even}, is an equivalence relation. Show that all the elements of {1, 3, 5} are related to each other and all the elements of {2, 4} are related to each other. But no element of {1, 3, 5} is related to any element of {2, 4}.

- Q:- Show that the relation R in the set R of real numbers, defined as R = {(a, b): a ≤ b
^{2}} is neither reflexive nor symmetric nor transitive. - Q:- Given 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. - Q:- Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as

R = {(a, b): b = a + 1} is reflexive, symmetric or transitive. - Q:-
Find gof and fog, if

(i)

*f(x)*= | x | and*g(x)*= | 5x – 2 |

(ii)*f(x)*= 8x3 and*g(x)*= x^{1/3}. - Q:- If a matrix has 24 elements, what are the possible order it can have? What, if it has 13 elements?
- Q:-
Show that the Modulus Function

*f*: R → R, given by f(x) = |x|, is neither oneone nor onto, where | x | is x, if x is positive or 0 and |x| is – x, if x is negative. - Q:- Show that each of the relation R in the set A = { x ∈Z: 0≤x≤12}, A={x} given by

(i) R = { (a,b) : |a - b| is a multiple of 4}

(ii) R = {(a,b):a = b} is an equivalence relation.

Find the set of all elements related to 1 in each case. - Q:-
Consider

*f*: R → R given by*f(x)*= 4x + 3. Show that*f*is invertible. Find the inverse of*f*. - Q:-
The radius of an air bubble is increasing at the rate of 1/2 cm/s. At what rate is the volume of the bubble increasing when the radius is 1 cm?

- Q:-
In Figure, identify the following vectors.

(i) Coinitial (ii) Equal (iii) Collinear but not equal

- NCERT Chapter

Copyright © 2020 saralstudy.com. All Rights Reserved.