Sample QuestionsRelations and Functions questions
One sample from each question group in this chapter. Select any group above to see the full set with answer keys.
The range of the function $\text{f(x)}=^{7-\text{x}}\text{P}_{\text{x}-3}$ is:
- A
$\{1,2,3,4,5\}$
- B
$\{1,2,3,4,3,6\}$
- C
$\{1,2,3,4\}$
- ✓
$\{1,2,3\}$
Answer: D.
View full solution →If $f: R \rightarrow R, g: R \rightarrow R$ and $h: R \rightarrow R$ are such that $f(x)=x^2, g(x)=\tan x$ and $h(x)=\log x$, then the value of $(go(foh)) ( x )$, if $x=1$ will be:
Answer: D.
View full solution →On the power set $P$ of a non$-$empty set $A,$ we define an operation $\triangle \text{ by }\text{X}\triangle\text{Y}=(\text{X}\cap\text{Y})∪(\text{X}∩\text{Y})\text{X}\triangle\text{Y}=\text{X}∩\text{Y}∪\text{X}∩\text{Y}$
Then which are of the following statements is true about $\triangle$
- A
Commutative and associative without an identity.
- B
Commutative but not associative with an identity.
- C
Associative but not commutative without an identity.
- ✓
Associative and commutative with an identity.
Answer: D.
View full solution →The relation $S$ defined on the set $R$ of all real number by the rule $aSb$ iff $a ≥ b$ is:
- A
- ✓
Reflexive, transitive but not symmetric.
- C
Symmetric, transitive but not reflexive.
- D
Neither transitive nor reflexive but symmetric.
Answer: B.
View full solution →$S$ is a relation over the set $R$ of all real numbers and it is given by $(\text{a, b})\in\text{S}\Leftrightarrow\text{ab}\geq0.$ Then$, S$ is:
- A
Symmetric and transitive only.
- B
Reflexive and symmetric only.
- C
- ✓
Answer: D.
View full solution →Directions: In the following questions, a statement of assertion $(A)$ is followed by a statement of reason $(R).$ Mark the correct choice as:
If $A = \{1, 2, 3\}, B = \{4,5, 6, 7\}$ and $f = \{(1, 4), (2,5), (3, 6)\}$ is a function from $A$ to $B.$
Assertion: $f(x)$ is a one $-$ one function.
Reason: $f(x)$ is an onto function.
- A
Both $A$ and $R$ are true and $R$ is the correct explanation of $A.$
- B
Both $A$ and $R$ are true but $R$ is not the correct explanation of $A.$
- ✓
$A$ is true but $R$ is false.
- D
$A$ is false and $R$ is true.
Answer: C.
View full solution →Directions: In the following questions, a statement of assertion $(A)$ is followed by a statement of reason $(R).$ Mark the correct choice as:
Assertion: A relation $R = \{(1, 1), (1, 2), (2, 2), (2, 3), (3, 3)\}$ defined on the set $A = \{1, 2, 3\}$ is symmetri.
Reason: A relation $R$ on the set $A$ is symmetric $(\text{a},\text{b})\in\text{R}$
$\Rightarrow(\text{b},\text{a})\in\text{R}.$
- A
Both $A$ and $R$ are true and $R$ is the correct explanation of $A.$
- B
Both $A$ and $R$ are true but $R$ is not the correct explanation of $A.$
- C
$A$ is true but $R$ is false.
- ✓
$A$ is false but $R$ is true.
Answer: D.
View full solution →Directions: In the following questions, a statement of assertion $(A)$ is followed by a statement of reason $(R).$ Mark the correct choice as:
Consider the set $A = \{1, 3, 5\}.$
Assertion: The number of reflexive relations on set $A$ is $2^9.$
Reason: A relation is said to be reflexive if xRx, $\forall\ \text{x}\in\text{A}.$
- A
Both $A$ and $R$ are true and $R$ is the correct explanation of $A.$
- B
Both $A$ and $R$ are true but $R$ is not the correct explanation of $A.$
- C
$A$ is true but $R$ is false.
- ✓
$A$ is false and $R$ is true.
Answer: D.
View full solution →Directions: In the following questions, a statement of assertion $(A)$ is followed by a statement of reason $(R).$ Mark the correct choice as:
Assertion: $A = \{1, 2, 3\}, B = \{4, 5, 6, 7\}, f = \{(1, 4), (2, 5), (3, 6)\}$ is a function from $A$ to $B.$Then $f$ is one $-$ one.
Reason: A function $f$ is one $-$ one if distinct elements of $A$ have distinct images in $B.$
- ✓
Both $A$ and $R$ are true and $R$ is the correct explanation of $A.$
- B
Both $A$ and $R$ are true but $R$ is not the correct explanation of $A.$
- C
$A$ is true but $R$ is false.
- D
$A$ is false but $R$ is true.
Answer: A.
View full solution →Directions: In the following questions, a statement of assertion $(A)$ is followed by a statement of reason $(R)$. Mark the correct choice as:
Assertion: If $n(A) = p$ and $n(B) = q$ then the number of relations from $A$ to $B$ is $2^{pq}.$
Reason: A relation from $A$ to $B$ is a subset of $A \times B.$
- ✓
Both $A$ and $R$ are true and $R$ is the correct explanation of $A.$
- B
Both $A$ and $R$ are true but $R$ is not the correct explanation of $A.$
- C
$A$ is true but $R$ is false.
- D
$A$ is false but $R$ is true.
Answer: A.
View full solution →Let $A = \{1, 2, 3\}$. Then number of equivalence relations containing $(1, 2)$ is
View full solution →Let A = {1, 2, 3}. Then number of relations containing (1, 2) and (1, 3) which are reflexive and symmetric but not transitive is
Find the number of all onto functions from the set $\{1, 2, 3, ...., n\}$ to itself.
View full solution →Show that the function $f : R \rightarrow R$ given by $f(x) = x^3$ is injective.
View full solution →Let $\;f{\text{ }}:{\text{ }}R{\text{ }} \to {\text{ }}R$ be defined as f (x) = 3x. Choose the correct answer.
View full solution →Given a non empty set X, consider P (X) which is the set of all subsets of X.
Define the relation R in P (X) as follows:
For subsets A, B in P (X), ARB if and only if A $\subset$ B. Is R an equivalence relation on P (X)? Justify your answer.
View full solution →Show that the function f : R $\rightarrow$ {x $\in$ R : -1 < x < 1} defined by $f(x) = \frac{x}{{1 + |x|}}$, x $\in$ R is one-one and onto function.
View full solution →Let $A$ and $B$ be sets. Show that $f : A \times B \rightarrow B \times A$ such that $f(a, b) = (b, a)$ is a bijective function.
View full solution →State whether the function is one-one, onto or bijective. Justify your answer. $f: R \rightarrow R$ defined by $f(x) = 1+ x^2$
View full solution →State whether the function is one-one, onto or bijective. Justify your answer. $f: R \rightarrow R$ defined by $f(x) = 3 - 4x.$
View full solution →Let $A =\{-1, 0, 1, 2\}, B = \{-4, -2, 0, 2\}$ and $f, g : A \rightarrow B $ be the functions defined by $f(x) = x^2 - x, x \in A$ and $g(x) = 2\left| {x - \frac{1}{2}} \right| - 1,x \in A$. Are $f$ and $g$ equal? Justify your answer.
(Hint: One may note that two functions $f : A \rightarrow B$ and $g : A \rightarrow B$ such that $f(a) = g(a)$ $\forall$ a $\in A,$ are called equal functions).
View full solution →If f: N $\to$ N is defined by f(n) = $\left\{ \begin{array} { l } { \frac { n + 1 } { 2 } , \text { if } n \text { is odd } } \\ { \frac { n } { 2 } , \text { if } n \text { is even } } \end{array} \right.$for all n $ \in$ N. State whether the function f is bijective. Justify your answer.
View full solution →Let $A = R - \{3\}$ and $B = R - \{1\}$. Consider the function $f : A \rightarrow B$ defined by $f(x) = \left( {\frac{{x - 2}}{{x - 3}}} \right)$ Is f one-one and onto? Justify your answer.
View full solution →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}.
Show that the relation R in the set A of all the books in a library of a college, given by R = {(x, y) : x and y have same number of pages} is an equivalence relation.
If A = { 1, 2, 3}, B = { 4, 5, 6, 7} and let f = {(1, 4), (2, 5), (3, 6)} be a function from A to B. Show that f is one-one.
A relation R on a set A is said to be an equivalence relation on A iff it is:
- Reflexive i.e., $(\text{a, a})\in\ \text{R} \ \forall \ \text{a}\in\text{A}.$
- Symmetric i.e., $(\text{a, b})\in\ \text{R} \Rightarrow \text{(b, a) } \in\text{R}\ \forall \ \text{a, b}\in\text{A}.$
- Transitive i.e., $(\text{a, b})\in\ \text{R} \ \text{and}\ \text{(b, c) } \in\text{R}\Rightarrow\text{(a, c)}\in\text{R}\ \forall \ \text{a, b, c}\in\text{A}.$
Based on the above information, answer the following questions.
- If the relation R = {(1, 1), (1, 2), (1, 3), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)} defined on the set A = {1, 2, 3}, then R is:
- Reflexive
- Symmetric
- Transitive
- Equivalence
- If the relation R = {(1, 2), (2, 1), (1, 3), (3, 1)} defined on the set A = {1, 2, 3}, then R is:
- Reflexive
- Symmetric
- Transitive
- Equivalence
- If the relation R on the set N of all natural numbers defined as R = {(x, y): y = x + 5 and x < 4}, then R is:
- Reflexive
- Symmetric
- Transitive
- Equivalence
- If the relation R on the set A = {1, 2, 3, ........., 13, 14} defined as R = {(x, y): 3x - y = O}, then R is:
- Reflexive
- Symmetric
- Transitive
- Equivalence
- If the relation R on the set A = {I, 2, 3} defined as R = {(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)}, then R is:
- Reflexive only
- Symmetric only
- Transitive only
- Equivalence
View full solution →Consider the mapping $f: A → B$ is defined by $f(x) = x - 1$ such that $f $ is a bijection.
Based on the above information, answer the following questions.
- Domain of $f$ is:
- $R - {2}$
- $R$
- $R - {1, 2}$
- $R - {0}$
- Range of $f$ is:
- $R$
- $R - {2}$
- $R - {0}$
- $R - {1, 2}$
- If $g: R - {2} → R - \{1\}$ is defined by $g(x) = 2f(x) - 1,$ then $g(x)$ in terms of $x$ is:
- $\frac{\text{x}+2}{\text{x}}$
- $\frac{\text{x}+1}{\text{x}-2}$
- $\frac{\text{x}-2}{\text{x}}$
- $\frac{\text{x}}{\text{x}-2}$
- The function $g$ defined above, is:
- One-one
- Many-one
- into
- None of these
- $A$ function $f(x)$ is said to be one-one iff.
- $f(x_1) = f(x_2) \Rightarrow -x_1= x_2$
- $f(-x_1) = f(-x_2) \Rightarrow -x_1 = x_2$
- $f(x_1) = f(x_2) \Rightarrow x_1 = x_2$
- None of these
View full solution →