Relations and Functions Test

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Relations and Functions Test - 2

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Weekly Quiz Competition

Question 1
1 / -0.25

If R is a relation from a non –empty set A to a non –empty set B, then

A
R ⊆A ×B.

B
R = A ∪B

C
R = A ∩B

D
R =A ×B

Solution

Let A and B be two sets. Then a relation R from set A to set B is a subset of A ×B. Thus, R is a relation from A to B ⇔R ⊆A ×B.
Question 2
1 / -0.25

The range of the function f(x) = ^7-x P_x-3 is

A
{1, 2, 3}

B
{1, 2, 3, 4, 5, 6}

C
{1, 2, 3, 4}

D
{1, 3}

Solution

Here, 0 ≤x- 3 ≤7 - x
⇒0 ≤x - 3 and x - 3 ≤7 - x
By solvation, we will get 3 ≤x ≤5
So x = 3,4,5 find the values of ^7-x P_{x - 3} by substituting the values of x
Question 3
1 / -0.25

Let R be the relation over the set of straight lines of a plane such that l₁ R l₂ ⇔l₁ ⊥l₂ . Then, R is

A
Symmetric

B
Reflexive

C
Transitive

D
An equivalence relation

Solution

To be reflexive, a line must be perpendicular to itself, but which is not true. So, R is not reflexive
For symmetric, if l₁ R l₂ ⇒l₁ ⊥l₂ .
⇒ l₂ ⊥l₁ ⇒l₁ R l₂ hence symmetric
For transitive, if l₁ R l₂ and l₂ R l₃
⇒l₁ R l₂ and l₂ R l₃ does not imply that l₁ ⊥l₃ hence not transitive.
Question 4
1 / -0.25

The diagram given below shows that

A
f is a one –one function from A to B

B
f is not a function

C
f is a function from A to B.

D
f is a bijection function from A to B

Solution

Because, the element b in the domain A has no image in the co-domain B.
Question 5
1 / -0.25

Which of the following is an even function?

A
x² + sin² x

B
sin³ xsin³ x

C
√x

D
None of these

Solution

Because, f(- x) = f(x) is the necessary condition for a function to be an even function, which is only satisfied by x² + sin² x .
Question 6
1 / -0.25

The binary relation S = Φ(empty set) on set A = {1, 2, 3} is

A
Transitive and reflexive

B
Symmetric and reflexive

C
Transitive and symmetric

D
Neither reflexive nor symmetric

Solution

Reflexive : A relation is reflexive if every element of set is paired with itself. Here none of the element of A is paired with themselves, so S is not reflexive.
Symmetric : This property says that if there is a pair (a, b) in S, then there must be a pair (b, a) in S. Since there is no pair here in S, this is trivially true, so S is symmetric.
Transitive : This says that if there are pairs (a, b) and (b, c) in S, then there must be pair (a,c) in S. Again, this condition is trivially true, so S is transitive.
Question 7
1 / -0.25

The void relation (a subset of A x A) on a non empty set A is:

A
Anti symmetric

B
Transitive

C
Reflexive

D
Transitive and symmetric

Solution

The relation { } ⊂ A x A on a is surely not reflexive. However, neither symmetry nor transitivity is contradicted. So { } is a transitive and symmetry relation on A.
Question 8
1 / -0.25

A relation R in a set A is called reflexive,

A
If (b, b) ∈R, for every a ∈A

B
If (b, a) ∈R, for every a, b ∈A

C
If (a, b) ∈R, for every a, b ∈A

D
If (a, a) ∈R, for every a ∈A

Solution

A relation R on a non empty set A is said to be reflexive if fx Rx for all x ∈ R, Therefore, R is reflexive.
Question 9
1 / -0.25

The domain of the function f = {(1, 3), (3, 5), (2, 6)} is

A
{1, 3, 2}

B
1, 3 and 2

C
3, 5 and 6

D
{3, 5, 6}

Solution

The domain in ordered pair (x,y) is represented by x-coordinate. Therefore, the domain of the given function is given by : {1, 3, 2}.
Question 10
1 / -0.25

The domain of the function

A
[0, ∞)

B
(1, 6)

C
[1, 6]

D
None of these.

Solution

x - 1 ≥0 and 6 –x ≥0 ⇒ 1 ≤x ≤6.

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Relations and Functions Test - 2

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Question 1

R ⊆A ×B.

R = A ∪B

R = A ∩B

R =A ×B

Solution

Question 2

{1, 2, 3}

{1, 2, 3, 4, 5, 6}

{1, 2, 3, 4}

{1, 3}

Solution

Question 3

Symmetric

Reflexive

Transitive

An equivalence relation

Solution

Question 4

f is a one –one function from A to B

f is not a function

f is a function from A to B.

f is a bijection function from A to B

Solution

Question 5

x2 + sin2 x

sin3 xsin3 x

√x

None of these

Solution

Question 6

Transitive and reflexive

Symmetric and reflexive

Transitive and symmetric

Neither reflexive nor symmetric

Solution

Question 7

Anti symmetric

Transitive

Reflexive

Transitive and symmetric

Solution

Question 8

If (b, b) ∈R, for every a ∈A

If (b, a) ∈R, for every a, b ∈A

If (a, b) ∈R, for every a, b ∈A

If (a, a) ∈R, for every a ∈A

Solution

Question 9

{1, 3, 2}

1, 3 and 2

3, 5 and 6

{3, 5, 6}

Solution

Question 10

[0, ∞)

(1, 6)

[1, 6]

None of these.

Solution

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x² + sin² x

sin³ xsin³ x