Relations and Functions Test

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Relations and Functions Test - 3

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Question 1
1 / -0.25

Let R be the relation on N defined as x R y if x + 2 y = 8. The domain of R is

A
{2, 4, 8}

B
{1, 2, 3, 4}

C
{2, 4, 6, 8}

D
{2, 4, 6}

Solution

As x R y if x + 2y = 8, therefore, domain of the relation R is given by x = 8 –2y ∈N.
When y = 1,
⇒x = 6 ,when y = 2,
⇒x = 4, when y = 3,
⇒x = 2.
therefore domain is {2, 4, 6}.
Question 2
1 / -0.25

If n ≥ 2, then the number of onto mappings or surjections that can be defined from {1, 2, 3, 4, ……….., n} onto {1, 2} is

A
2n

B
2ⁿ − 2

C
2ⁿ + 2

D
2ⁿ

Solution

The number of onto functions that can be defined from a finite set A containing n elements onto a finite set B containing 2 elements = 2ⁿ − 2.
Question 3
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A relation R in a set A is called symmetric, if

A
(a₁ , a₂ ) ∈R implies that (a₂ , a₂ ) ∈R, for all a₁ , a₂ ∈A

B
(a₂ , a₂ ) ∈R implies that (a₂ , a₁ ) ∈R, for all a₁ , a₂ ∈A

C
(a₁ , a₂ ) ∈R implies that (a₂ , a₁ ) ∈R, for all a₁ , a₂ ∈A

D
(a₁ , a₁ ) ∈R implies that (a₂ , a₁ ) ∈R, for all a₁ , a₂ ∈A

Solution

A relation R on a non empty set A is said to be symmetric if fx Ry ⇔yRx, for all x , y ∈R .
Question 4
1 / -0.25

The range of is

A
–1

B
{1}

C
{–1}

D
1

Solution

We have ,

Therefore, range of f(x) is {-1}.
Question 5
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The function f(x) = sin x² is

A
Periodic

B
An even function

C
One –one

D
None of these.

Solution

For even function: f(-x) = f(x) ,
therefore, f(− x)
= sin (− x)² = sin x² = f(x).
Question 6
1 / -0.25

Which of the following is not an equivalence relation on I, the set of integers ; x, y

A
x R y if x + y is an even integer

B
x R y if x = y

C
x R y if x

D
x R y if x –y is an even integer

Solution

If R is a relation defined by xRy : ifx ⩽y, then R is reflexive and transitive But, it is not symmetric. Hence, R is not an equivalence relation.
Question 7
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If A = {1, 2, 3}, then the relation R = {(1, 2), (2, 3), (1, 3) in A is

A
Symmetric only

B
Symmetric and transitive only

C
Transitive only

D
None of these

Solution

A relation R on a non empty set A is said to be transitive if fxRy and y Rz ⇒xRz, for all x ∈ R. Here, (1, 2) and (2, 3) belongs to R implies that (1, 3) belongs to R.
Question 8
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A relation R in a set A is called transitive, if

A
(a₁ , a₂ ) ∈R implies that (a₁ , a₃ ) ∈R, for all a₁ , a₂ , (a₁ , a₃ )

B
∈R and (a₂ , a₃ ) ∈R implies that (a₁ , a₂ ) ∈R, for all a₁ , a₂ ,

C
(a₁ , a₂ ) ∈R and (a₂ , a₃ ) ∈R implies that (a₁ , a₃ ) ∈R, for all a₁ , a₂ ,

D
(a_1, a₁ ) ∈R and (a₂ , a₂ ) ∈R implies that (a₁ , a₃ ) ∈R, for all a₁ , a₂ ,

Solution

A relation R on a non empty set A is said to be transitive if fx Ry and yRz ⇒x Rz, for all x ∈ R.
Question 9
1 / -0.25

The range of the function f(x) =|x −1| is

A
[0, ∞)

B
(−∞, 0)

C
(−∞, ∞)

D
(0, ∞)

Solution

We have, f(x) = |x −1|, which always gives non-negative values of f(x) for all x ∈R.Therefore range of the given function is all non-negative real numbers i.e. [0,∞).
Question 10
1 / -0.25

The range of the function is

A
{1}

B
{0, 1, –1}.

C
{–1}

D
{1, –1}

Solution

As the denominator of the function is a modulus function i.e.
Question 11
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Let A = {a, b, c} and R = {(a, a), (b, b), (c, c), (b, c)} be a relation on A. Here, R is

A
Transitive

B
Reflexive

C
Symmetric

D
None

Solution

Correct Answer :- b

Explanation:- A = {a, b, c} and R = {(a, a), (b, b), (c, c), (b, c)}

Any relation R is reflexive if fx Rx for all x ∈R. Here ,(a, a), (b, b), (c, c) ∈R. Therefore , R is reflexive.

For the transitive, in the relation R there should be (a,c)

Hence it is not transitive.
Question 12
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A relation R from C to R is defined by x Ry iff |x| = y. Which of the following is correct?

A
iR1

B
3R(–3)

C
(2 + 3 i)R13

D
(1 + i)R2

Solution
Question 13
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A relation R in a set A is said to be an equivalence relation if

A
If R is reflexiveand transitive

B
If R is reflexive and symmetric

C
If R is reflexive, symmetric and transitive

D
If R is symmetric and transitive

Solution

A relation R on a non empty set A is said to be reflexive iff xRx for all x ∈ R . .
A relation R on a non empty set A is said to be symmetric if fx Ry ⇔y Rx, for all x , y ∈R .
A relation R on a non empty set A is said to be transitive if fx Ry and y Rz ⇒x Rz, for all x ∈ R.
An equivalence relation satisfies all these three properties.
Question 14
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Let f: R →R be a mapping such that f(x) = . Then f is

A
Many –one and onto

B
One –one and onto

C
Into and one one

D
Many one and into

Solution

Correct answer is D.
Question 15
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Which of the following is a polynomial function?

A
x³ +3x² −4x+2 –√2x⁻²

B

C

D

Solution

A polynomial function has all exponents as integral whole numbers.