Relation and Function Set-1

Question 1

Let $T$ be the set of all triangles in the Euclidean plane, and let a relation $R$ on $T$ be defined as $a R b$ if $a$ is congruent to $b$, $a, b \in T$. Then $R$ is

Accepted Answer

equivalence

Answer

reflexive but not transitive

Answer

transitive but not symmetric

Answer

none of these

Question 2

The maximum number of equivalence relations on the set $A = \{2, 3, 4\}$ are

Accepted Answer

5

Answer

1

Answer

27

Answer

3

Question 3

If $f : R \rightarrow R$ be the function defined by $f(x) = x^3 + 5$, then $f^{–1}(x)$ is

Accepted Answer

$(x – 5)^{1/3}$

Answer

$(x + 5)^{1/3}$

Answer

$(5 – x)^{1/3}$

Answer

$(5 – x)$

Question 4

If a relation $R$ on the set $\{1, 2, 3\}$ be defined by $R = \{(1, 2)\}$, then $R$ is

Accepted Answer

transitive

Answer

reflexive

Answer

symmetric

Answer

none of these

Question 5

If $f : R \rightarrow R$ be defined by $f(x) = 2/x$, $x \forall R$, then $f$ is

Accepted Answer

$f$ is not defined

Answer

one-one

Answer

onto

Answer

bijective

Question 6

If $f : A \rightarrow B$ and $g : B \rightarrow C$ be the bijective functions, then $(gof)^{–1}$ is

Accepted Answer

$f^{–1}og^{–1}$

Answer

$fog$

Answer

$g^{–1}of^{–1}$

Answer

$gof$

Question 7

Which of the following functions form $Z$ into $Z$ bijections?

Accepted Answer

$f (x) = x + 2$

Answer

$f (x) = x^3$

Answer

$f (x) = 2x + 1$

Answer

$f (x) = x^2 + 1$

Question 8

If the set $A$ contains 7 elements and the set $B$ contains 8 elements, then number of one-one and onto mappings from $A$ to $B$ is

Accepted Answer

0

Answer

24

Answer

120

Answer

none of these

Question 9

If $f : R – \{3/5\} \rightarrow R$ be defined by $f (x) = \frac{3x + 2}{5x - 3}$ then

Accepted Answer

$f^{–1}(x) = f (x)$

Answer

$f^{–1}(x) = –f (x)$

Answer

$fof (x) = –x$

Answer

$f^{–1}(x) = \frac{1}{19} f (x)$

Question 10

Let $A = \{1, 2, 3, 4\}$. Let $R$ be the equivalence relation on $A 	imes A$ defined by $(a, b) R (c, d)$ if $a + d = b + c$. Then the equivalence class $[(1, 3)]$ is

Accepted Answer

$\{(1, 3), (2, 4)\}$

Answer

$\{(1, 3)\}$

Answer

$\{(2, 4)\}$

Answer

$\{(1, 8), (2, 4), (1, 4)\}$

Question 11

Let $f : N \rightarrow R$ be the function defined by $f (x) = \frac{2x - 1}{2}$ and $g : Q \rightarrow R$ be another function defined by $g (x) = x + 2$. Then $(gof) \frac{3}{2}$ is

Accepted Answer

3

Answer

1

Answer

– 1

Answer

$7/2$

Question 12

If $f: R\rightarrow R$ given by $f(x) =(3 − x^3)^{1/3}$, find $fof(x)$

Accepted Answer

$x$

Answer

$(3- x^3)$

Answer

$x^3$

Answer

None of these

Question 13

Let $A = \{1,2,3\}$. The number of equivalence relations containing $(1,2)$ is

Accepted Answer

2

Answer

3

Answer

4

Answer

None of these

Question 14

Let $f:R\rightarrow R$ defined by $f(x) = x^4$. Choose the correct answer

Accepted Answer

$f$ is neither one-one nor onto

Answer

$f$ is one-one but not onto

Answer

$f$ is many one onto

Answer

None of these

Question 15

Let $f:R\rightarrow R$ defined by $f(x) = 3x$. Choose the correct answer

Accepted Answer

$f$ is one one onto

Answer

$f$ is many one onto

Answer

$f$ is one-one but not onto

Answer

$f$ is neither one-one nor onto

Question 16

If $A = \{1,2,3\}$, $B = \{4,6,9\}$ and $R$ is a relation from $A$ to $B$ defined by ‘$x$ is smaller than $y$’. The range of $R$ is

Accepted Answer

$\{4,6,9\}$

Answer

$\{1\}$

Answer

none of these

Answer

$\{1, 4,6,9\}$

Question 17

The relation $R = \{ (1,1),(2,2),(3,3)\}$ on $\{1,2,3\}$ is

Accepted Answer

an equivalence relation

Answer

transitive only

Answer

reflexive only

Answer

None of these

Question 18

Let $A =\{1,2,3\}$ and consider the relation $R= \{(1,1),(2,2),(3,3),(1,2),(2,3),(1,3)\}$ then $R$ is

Accepted Answer

reflexive but not symmetric

Answer

symmetric and transitive

Answer

reflexive but not transitive

Answer

neither symmetric nor transitive

Question 19

Let us define a relation $R$ in $R$ as $a R b$ if $a\geq b$. Then $R$ is

Accepted Answer

reflexive, transitive but not symmetric

Answer

neither transitive nor reflexive but

Answer

an equivalence relation

Answer

symmetric ,transitive but not reflexive

Question 20

Domain of $f$ is (Context: $f(x) = \frac{x − 1}{x – 2}$)

Accepted Answer

$R – \{2\}$

Answer

$R$

Answer

$R – \{1, 2\}$

Answer

$R – \{0\}$

Question 21

If $g : R – \{2\} \rightarrow R – \{1\}$ is defined by $g(x) = 2f(x) – 1$, then $g(x)$ in terms of $x$ is (Context: $f(x) = \frac{x − 1}{x – 2}$)

Accepted Answer

$\frac{x}{x – 2}$

Answer

$\frac{x + 2}{x}$

Answer

$\frac{x + 1}{x – 2}$

Answer

$\frac{x − 2}{x}$

Question 22

A function $f(x)$ is said to be one-one if

Accepted Answer

$f(x_1) = f(x_2) \Rightarrow x_1 = x_2$

Answer

$f(x_1) = f(x_2) \Rightarrow – x_1 = x_2$

Answer

$f(–x_1) = f(–x_2) \Rightarrow– x_1= x_2$

Answer

None of these

Question 23

Range of $f$ is (Context: $f(x) = \frac{x − 1}{x – 2}$)

Accepted Answer

$R – \{1\}$

Answer

$R$

Answer

$R – \{0\}$

Answer

$R – \{1, 2\}$

Question 24

The function $f : A \rightarrow B$ defined by $f(x) = 4x + 7, x \in R$ is

Accepted Answer

one-one

Answer

Many-one

Answer

Odd

Answer

Even

Question 25

The smallest integer function $f(x) = [x]$ is

Accepted Answer

Many-one

Answer

One-one

Answer

Both (a) & (b)

Answer

None of these

Question 26

The function $f : R \rightarrow R$ defined by $f(x) = 3 – 4x$ is

Accepted Answer

Onto

Answer

Not onto

Answer

None one-one

Answer

None of these

Question 27

The number of bijective functions from set $A$ to itself when $A$ contains $10^6$ elements is

Accepted Answer

$10^6!$

Answer

$10^6$

Answer

$(10^6)^2$

Answer

$2^{10^6}$

Question 28

If $f(x) = (ax^2 + b)^3$, then the function $g$ such that $f(g(x)) = g(f(x))$ is given by

Accepted Answer

$g(x)=\left(\frac{x^{1 / 3}-b}{a}\right)^{1 / 2}$

Answer

$g(x)=\left(\frac{b-x^{1 / 3}}{a}\right)$

Answer

$g(x)=\frac{1}{\left(a x^{2}+b\right)^{3}}$

Answer

$g(x)=\left(a x^{2}+b\right)^{1 / 3}$

Question 29

If $f : R ightarrow R$, $g : R ightarrow R$ and $h : R ightarrow R$ is such that $f(x) = x^2$, $g(x) = 	an x$ and $h(x) = \log x$, then the value of $[ho(gof)](x)$, if $x = \frac{\sqrt{\pi}}{2}$ will be

Accepted Answer

0

Answer

1

Answer

-1

Answer

10

Question 30

If $f : R \rightarrow R$ and $g : R \rightarrow R$ defined by $f(x) = 2x + 3$ and $g(x) = x^2 + 7$, then the value of $x$ for which $f(g(x)) = 25$ is

Accepted Answer

$\pm 2$

Answer

$\pm 1$

Answer

$\pm 3$

Answer

$\pm 4$

Question 31

If $f(x)=\frac{x-1}{x+1}$, then $f(f(x))$ is

Accepted Answer

$-\frac{1}{x}$

Answer

$\frac{1}{x}$

Answer

$\frac{1}{x+1}$

Answer

$\frac{1}{x-1}$

Question 32

If $f(x) = 1-\frac{1}{x}$, then $f(f(\frac{1}{x}))$

Accepted Answer

$\frac{x}{x-1}$

Answer

$\frac{1}{x}$

Answer

$\frac{1}{1+x}$

Answer

$\frac{1}{x-1}$

Question 33

If $f : R ightarrow R, g : R ightarrow R$ and $h : R ightarrow R$ are such that $f(x) = x^2, g(x) = 	an x$ and $h(x) = \log x$, then the value of $(go(foh)) (x)$, if $x = 1$ will be

Accepted Answer

0

Answer

1

Answer

-1

Answer

$\pi$

Question 34

If $f(x) = \frac{3 x+2}{5 x-3}$ then $(fof)(x)$ is

Accepted Answer

$x$

Answer

$-x$

Answer

$f(x)$

Answer

$-f(x)$

Question 35

If the binary operation $*$ is defind on the set $Q^+$ of all positive rational numbers by $a * b = \frac{a b}{4}$. Then, $3 *(\frac{1}{5} * \frac{1}{2})$ is equal to

Accepted Answer

$\frac{3}{160}$

Answer

$\frac{5}{160}$

Answer

$\frac{3}{10}$

Answer

$\frac{3}{40}$

Question 36

The number of binary operations that can be defined on a set of 2 elements is

Accepted Answer

16

Answer

8

Answer

4

Answer

64

Question 37

Let $*$ be a binary operation on $Q$, defined by $a * b = \frac{3 a b}{5}$ is

Accepted Answer

Both (a) and (b)

Answer

Commutative

Answer

Associative

Answer

None of these

Question 38

Let $*$ be a binary operation on set $Q$ of rational numbers defined as $a * b = \frac{a b}{5}$. Write the identity for $*$.

Accepted Answer

5

Answer

3

Answer

1

Answer

6

Question 39

For binary operation $*$ defind on $R – \{1\}$ such that $a * b = \frac{a}{b+1}$ is

Accepted Answer

both (a) and (b)

Answer

not associative

Answer

not commutative

Answer

commutative

Question 40

The binary operation $*$ defind on set $R$, given by $a * b = \frac{a+b}{2}$ for all $a,b \in R$ is

Accepted Answer

commutative

Answer

associative

Answer

Both (a) and (b)

Answer

None of these

Question 41

Let $A = N 	imes N$ and $*$ be the binary operation on $A$ defined by $(a, b) * (c, d) = (a + c, b + d)$. Then $*$ is

Accepted Answer

Both (a) and (b)

Answer

commutative

Answer

associative

Answer

None of these

Question 42

Find the identity element in the set $I^+$ of all positive integers defined by $a * b = a + b$ for all $a, b \in I^+$.

Accepted Answer

0

Answer

1

Answer

2

Answer

3

Question 43

Let $*$ be a binary operation on set $Q – \{1\}$ defind by $a * b = a + b – ab : a, b \in Q – \{1\}$. Then $*$ is

Accepted Answer

Both (a) and (b)

Answer

Commutative

Answer

Associative

Answer

None of these

Question 44

The binary operation $*$ defined on $N$ by $a * b = a + b + ab$ for all $a, b \in N$ is

Accepted Answer

both commutative and associative

Answer

commutative only

Answer

associative only

Answer

none of these

Question 45

The number of commutative binary operation that can be defined on a set of 2 elements is

Accepted Answer

2

Answer

8

Answer

6

Answer

4

Question 46

Let $T$ be the set of all triangles in the Euclidean plane, and let a relation $R$ on $T$ be defined as $a R b$ if $a$ is congruent to $b \forall a, b \in T$. Then $R$ is

Accepted Answer

equivalence

Answer

reflexive but not transitive

Answer

transitive but not symmetric

Answer

None of these

Question 47

The maximum number of equivalence relations on the set $A = \{1, 2, 3\}$ are

Accepted Answer

5

Answer

1

Answer

2

Answer

3

Question 48

Let us define a relation $R$ in $R$ as $a R b$ if $a \geq b$. Then $R$ is

Accepted Answer

reflexive, transitive but not symmetric

Answer

an equivalence relation

Answer

symmetric, transitive but not reflexive

Answer

neither transitive nor reflexive but symmetric

Question 49

Let $A = \{1, 2, 3\}$ and consider the relation $R = \{(1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (1, 3)\}$. Then $R$ is

Accepted Answer

reflexive but not symmetric

Answer

reflexive but not transitive

Answer

symmetric and transitive

Answer

neither symmetric, nor transitive

Question 50

The identity element for the binary operation $*$ defined on $Q – \{0\}$ as $a * b = \frac{a b}{2}$ $\forall a, b \in Q – \{0\}$ is

Accepted Answer

2

Answer

1

Answer

0

Answer

None of these

Question 51

Let $A = \{1, 2, 3, \ldots n\}$ and $B = \{a, b\}$. Then the number of surjections from $A$ into $B$ is

Accepted Answer

$2^n – 2$

Answer

$^n P_2$

Answer

$2^n – 1$

Answer

none of these

Question 52

Let $f : R \rightarrow R$ be defind by $f(x) = \frac{1}{x}$ $\forall x \in R$. Then $f$ is

Accepted Answer

$f$ is not defined

Answer

one-one

Answer

onto

Answer

bijective

Question 53

If $f : R ightarrow R$ be given by $f(x) = 	an x$. Then $f^{-1}(1)$ is

Accepted Answer

$\{n\pi + \frac{\pi}{4}; n \in Z\}$

Answer

$\frac{\pi}{4}$

Answer

Does not exist

Answer

None of these

Question 54

Let $R$ be a relation on the set $N$ of natural numbers denoted by $nRm \Leftrightarrow n$ is a factor of $m$ (i.e. $n | m$). Then, $R$ is

Accepted Answer

Reflexive, transitive but not symmetric

Answer

Reflexive and symmetric

Answer

Transitive and symmetric

Answer

Equivalence

Question 55

Let $S = \{1, 2, 3, 4, 5\}$ and let $A = S 	imes S$. Define the relation $R$ on $A$ as follows: $(a, b) R (c, d)$ iff $ad = cb$. Then, $R$ is

Accepted Answer

Equivalence relation

Answer

reflexive only

Answer

Symmetric only

Answer

Transitive only

Question 56

Total number of equivalence relations defined in the set $S = \{a, b, c\}$ is

Accepted Answer

5

Answer

$3!$

Answer

$2^3$

Answer

$3^3$

Question 57

The relation $R$ is defined on the set of natural numbers as $\{(a, b) : a = 2b\}$. Then, $R^{-1}$ is given by

Accepted Answer

$\{(1, 2), (2, 4), (3, 6), \ldots.\}$

Answer

$\{(2, 1), (4, 2), (6, 3),\ldots.\}$

Answer

$R^{-1}$ is not defiend

Answer

None of these

Question 58

Let $X = \{-1, 0, 1\}$, $Y = \{0, 2\}$ and a function $f : X \rightarrow Y$ defiend by $y = 2x^4$, is

Accepted Answer

many-one onto

Answer

one-one onto

Answer

one-one into

Answer

many-one into

Question 59

Let $A = R – \{3\}$, $B = R – \{1\}$. Let $f : A \rightarrow B$ be defined by $f(x)=\frac{x-2}{x-3}$. Then,

Accepted Answer

$f$ is bijective

Answer

$f$ is one-one but not onto

Answer

$f$ is onto but not one-one

Answer

None of these

Question 60

The mapping $f : N \rightarrow N$ is given by $f(n) = 1 + n^2, n \in N$ when $N$ is the set of natural numbers is

Accepted Answer

one-one but not onto

Answer

one-one and onto

Answer

onto but not one-one

Answer

neither one-one nor onto

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