Let $A=\{a, b, c\}$ and the relation $R$ be defined on $A$ as follows

$$R=\{(a, a),(b, c),(a, b)\}$$

Then, write minimum number of ordered pairs to be added in $R$ to make $R$ reflexive and transitive.

Let $D$ be the domain of the real valued function $f$ defined by $f(x)=\sqrt{25-x^2}$. Then, write $D$.

If $f, g: R \rightarrow R$ be defined by $f(x)=2 x+1$ and $g(x)=x^2-2, \forall x \in R$, respectively. Then, find $gof$.

Let $f: R \rightarrow R$ be the function defined by $f(x)=2 x-3, \forall x \in R$. Write $f^{-1}$.

If $A=\{a, b, c, d\}$ and the function $f=\{(a, b),(b, d),(c, a),(d, c)\}$, write $f^{-1}$.

If $f: R \rightarrow R$ is defined by $f(x)=x^2-3 x+2$, write $f\{f(x)\}$.

Is $g=\{(1,1),(2,3),(3,5),(4,7)\}$ a function? If $g$ is described by $g(x)=\alpha x+\beta$, then what value should be assigned to $\alpha$ and $\beta$ ?

Are the following set of ordered pairs functions? If so examine whether the mapping is injective or surjective.

(i) $\{(x, y): x$ is a person, $y$ is the mother of $x\}$.

(ii) $\{(a, b): a$ is a person, $b$ is an ancestor of $a\}$.

If the mappings $f$ and $g$ are given by $f=\{(1,2),(3,5),(4,1)\}$ and $g=\{(2,3),(5,1),(1,3)\}$, write $fog$.

Let $C$ be the set of complex numbers. Prove that the mapping $f: C \rightarrow R$ given by $f(z)=|z|, \forall z \in C$, is neither one-one nor onto.

Let the function $f: R \rightarrow R$ be defined by $f(x)=\cos x, \forall x \in R$. Show that $f$ is neither one-one nor onto.

Let $X=\{1,2,3\}$ and $Y=\{4,5\}$. Find whether the following subsets of $X \times Y$ are functions from $X$ to $Y$ or not.

(i) $f=\{(1,4),(1,5),(2,4),(3,5)\}$

(ii) $g=\{(1,4),(2,4),(3,4)\}$

(iii) $h=\{(1,4),(2,5),(3,5)\}$

(iv) $k=\{(1,4),(2,5)\}$

If functions $f: A \rightarrow B$ and $g: B \rightarrow A$ satisfy $g \circ f=I_A$, then show that $f$ is one-one and $g$ is onto.

Let $f: R \rightarrow R$ be the function defined by $f(x)=\frac{1}{2-\cos x}, \forall x \in R$. Then, find the range of $f$.

Let $n$ be a fixed positive integer. Define a relation $R$ in $Z$ as follows $\forall a$, $b \in Z, a R b$ if and only if $a-b$ is divisible by $n$. Show that $R$ is an equivalence relation.

If $A=\{1,2,3,4\}$, define relations on $A$ which have properties of being

(i) reflexive, transitive but not symmetric.

(ii) symmetric but neither reflexive nor transitive.

(iii) reflexive, symmetric and transitive.

Let $R$ be relation defined on the set of natural number $N$ as follows, $R=\{(x, y): x \in N, y \in N, 2 x+y=41\}$. Find the domain and range of the relation $R$. Also verify whether $R$ is reflexive, symmetric and transitive.

Given, $A=\{2,3,4\}, B=\{2,5,6,7\}$. Construct an example of each of the following

(i) an injective mapping from $A$ to $B$.

(ii) a mapping from $A$ to $B$ which is not injective.

(iii) a mapping from $B$ to $A$.

Give an example of a map

(i) which is one-one but not onto.

(ii) which is not one-one but onto.

(iii) which is neither one-one nor onto.

Let $A=R-\{3\}, B=R-\{1\}$. If $f: A \rightarrow B$ be defined by $f(x)=\frac{x-2}{x-3}$, $\forall x \in A$. Then, show that $f$ is bijective.

Let $A=[-1,1]$, then, discuss whether the following functions defined on $A$ are one-one onto or bijective.

(i) $f(x)=\frac{x}{2}\quad$ (ii) $g(x)=|x|$

(iii) $h(x)=x|x|\quad$ (iv) $k(x)=x^2$

Each of the following defines a relation of $N$

(i) $x$ is greater than $y, x, y \in N$.

(ii) $x+y=10, x, y \in N$.

(iii) $x y$ is square of an integer $x, y \in N$.

(iv) $x+4 y=10, x, y \in N$

Determine which of the above relations are reflexive, symmetric and transitive.

Let $A=\{1,2,3, \ldots, 9\}$ and $R$ be the relation in $A \times A$ defined by $(a, b) R(c, d)$ if $a+d=b+c$ for $(a, b),(c, d)$ in $A \times A$. Prove that $R$ is an equivalence relation and also obtain the equivalent class $[(2,5)]$.

Using the definition, prove that the function $f: A \rightarrow B$ is invertible if and only if $f$ is both one-one and onto.

Functions $f, g: R \rightarrow R$ are defined, respectively, by $f(x)=x^2+3 x+1$, $g(x)=2 x-3$, find

(i) $f \circ g$ (ii) $g \circ f$ (iii) $f \circ f$ (iv) $g \circ g$

Let * be the binary operation defined on $Q$. Find which of the following binary operations are commutative

(i) $a * b=a-b, \forall a, b \in Q$

(ii) $a * b=a^2+b^2, \forall a, b \in Q$

(iii) $a * b=a+a b, \forall a, b \in Q$

(iv) $a * b=(a-b)^2, \forall a, b \in Q$

If * be binary operation defined on $R$ by $a * b=1+a b, \forall a, b \in R$. Then, the operation $*$ is

(i) commutative but not associative.

(ii) associative but not commutative.

(iii) neither commutative nor associative.

(iv) both commutative and associative.

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

Consider the non-empty set consisting of children in a family and a relation $R$ defined as $a R b$, if $a$ is brother of $b$. Then, $R$ is

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

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

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

If $A=\{1,2,3\}$ and consider the relation

$$R=\{(1,1),(2,2),(3,3),(1,2),(2,3),(1,3)\}$$

Then, $R$ is

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

If the set $A$ contains 5 elements and the set $B$ contains 6 elements, then the number of one-one and onto mappings from $A$ to $B$ is

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

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

If $f: R \rightarrow R$ be defined by $f(x)=3 x^2-5$ and $g: R \rightarrow R$ by $g(x)=\frac{x}{x^2+1}$. Then, $g \circ f$ is

Which of the following functions from $Z$ into $Z$ are bijections?

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

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

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

If $f:[0,1] \rightarrow[0,1]$ be defined by $f(x)=\left\{\begin{array}{cc}x, & \text { if } x \text { is rational } \\ 1-x, & \text { if } x \text { is irrational }\end{array}\right.$ then $(f \circ f) x$ is

If $f:[2, \infty) \rightarrow R$ be the function defined by $f(x)=x^2-4 x+5$, then the range of $f$ is

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

If $f: R \rightarrow R$ be defined by $f(x)=\left\{\begin{array}{l}2 x: x>3 \\ x^2: 1< x \leq 3 \\ 3 x: x \leq 1\end{array}\right.$ Then, $f(-1)+f(2)+f(4)$ is

If $f: R \rightarrow R$ be given by $f(x)=\tan x$, then $f^{-1}(1)$ is

Let the relation $R$ be defined in $N$ by $a R b$, if $2 a+3 b=30$. Then, $R=$ ..........

If the relation $R$ be defined on the set $A=\{1,2,3,4,5\}$ by $R=\left\{(a, b):\left|a^2-b^2\right|< 8\right\}$. Then, $R$ is given by ............ .

If $f=\{(1,2),(3,5),(4,1)\}$ and $g=\{(2,3),(5,1),(1,3)\}$, then $g \circ f=\ldots \ldots \ldots$ and $f \circ g=\ldots \ldots \ldots$.

If $f: R \rightarrow R$ be defined by $f(x)=\frac{x}{\sqrt{1+x^2}}$, then $(f \circ f \circ f)(x)=$ .............. .

If $f(x)=\left[4-(x-7)^3\right]$, then $f^{-1}(x)=$ ............ .

Let $R=\{(3,1),(1,3),(3,3)\}$ be a relation defined on the set $A=\{1,2,3\}$. Then, $R$ is symmetric, transitive but not reflexive.

If $f: R \rightarrow R$ be the function defined by $f(x)=\sin (3 x+2) \forall x \in R$. Then, $f$ is invertible.

Every relation which is symmetric and transitive is also reflexive.

An integer $m$ is said to be related to another integer $n$, if $m$ is a integral multiple of $n$. This relation in $Z$ is reflexive, symmetric and transitive.

If $A=\{0,1\}$ and $N$ be the set of natural numbers. Then, the mapping $f: N \rightarrow A$ defined by $f(2 n-1)=0, f(2 n)=1, \forall n \in N$, is onto.

The relation $R$ on the set $A=\{1,2,3\}$ defined as $R=\{(1,1),(1,2),(2$, $1),(3,3)\}$ is reflexive, symmetric and transitive.

The composition of function is commutative.

The composition of function is associative.

Every function is invertible.

A binary operation on a set has always the identity element.