Question 1 :
$N$ is the set of positive integers. The relation $R$ is defined on N x N as follows: $(a,b) R (c,d)\Longleftrightarrow ad=bc$ Prove that
Question 2 :
The relation $R=\left\{ \left( 1,1 \right) ,\left( 2,2 \right) \left( 3,3 \right)  \right\} $ on the set $A=\left\{ 1,2,3 \right\} $ is
Question 5 :
Let $A=\left\{ 1,2,3 \right\} $ and $R=\left\{ \left( 1,2 \right) ,\left( 2,3 \right) ,\left( 1,3 \right) \right\} $ be a relation on $A$. Then $R$ is
Question 7 :
The relation $R$ in $N\times N$ such that $(a,b)R(c,d)\Leftrightarrow a+d=b+c$ is
Question 8 :
The true set of real value of $x$ for which the function, $f(x)=x\ \mathrm{ln}\ x-x+1$ is positive is
Question 9 :
In the set $Z$ of all integers, which of the following relation $R$ is an equivalence relation?
Question 10 :
Given the relation R= {(1,2), (2,3) } on the set {1, 2, 3}, the minimum number of ordered pairs which when added to R make it an equivalence relation is
Question 11 :
Let $L$ denote the set of all straight lines in a plane, Let a relation $R$ be defined by $lRm$, iff $l$ is perpendicular to $m$ for all $l \in L$. Then, $R$ is
Question 12 :
The relation $R$ on the set $Z$ of all integer numbers defined by $(x,y)\ \epsilon \ R\\Leftrightarrow x-y$ is divisible by $n$ is
Question 13 :
Let E = {1, 2, 3, 4} and F {1, 2}. Then the number of onto functions from E to F is
Question 14 :
Let $A=\left\{ 2,3,4,5,....,17,18 \right\} $. Let $\simeq $ be the equivalence relation on $A\times A$, cartesian product of $A$ with itself, defined by $(a,b)\simeq (c,d)$, iff $ad=bc$. The the number of ordered pairs of the equivalence class of $(3,2)$ is
Question 15 :
Let A={ 1, 2, 3, 4} and R= {( 2, 2), (3, 3), (4, 4), (1, 2)} be a relation on A. Then R is
Question 16 :
If the relation is defined on $R-\left\{ 0 \right\} $ by $\left( x,y \right) \in S\Leftrightarrow xy>0$, then $S$ is ________
Question 17 :
Consider the following two binary relations on the set $A = \left \{a, b, c\right \} : R_{1} = \left \{(c, a), (b, b), (a, c), (c, c), (b, c), (a, a)\right \}$<br>and $R_{2} = \left \{(a, b), (b, a), (c, c), (c, a), (a, a), (b, b), (a, c)\right \}$ Then<br>
Question 18 :
Let $f: N\rightarrow R$ such that $f(x)=\dfrac{2x-1}{2}$ and $g: Q\rightarrow R$such that $g(x)=x+2$ be two function. Then $(gof)\left(\dfrac{3}{2}\right)$ is equal to
Question 19 :
For real number $x$ and $y$, define $xRy$ iff $x-y+\sqrt{2}$ is an irrational number. Then the relation $R$ is
Question 20 :
Which one of the following relations on R (set of real numbers) is an equivalence relation
Question 21 :
Let $f(x,y)=xy^{2}$ if $x$ and $y$ satisfy $x^{2}+y^{2}=9$ then the minimum value of $f(x,y)$ is
Question 22 :
The number of reflexive relations of a set with four elements is equal to
Question 24 :
Let $f:R\rightarrow R$ be defined as $f(x)=x^{3}+2x^{2}+4x+\sin \left(\dfrac{\pi}{2}\right)$ and $g(x)$ be the inverse function of $f(x)$, then $g'(8)$ is equal to
Question 25 :
If $A=\left\{ a,b,c,d \right\} $, then a relation $R=\left\{ \left( a,b \right) ,\left( b,a \right) ,\left( a,a \right) \right\} $ on $A$ is
Question 26 :
Find number of all such functions $y = f(x)$ which are one-one?
Question 27 :
The number of reflexive relation in set A = {a, b, c} is equal to
Question 28 :
Let N denote the set of all natural numbers and R a relation on $N\times N$. Which of the following is an equivalence relation?
Question 29 :
Let $A = \left\{ {1,2,3} \right\}$ and $R = \left\{ {\left( {1,1} \right),\left( {1,3} \right),\left( {3,1} \right),\left( {2,2} \right),\left( {2,1} \right),\left( {3,3} \right)} \right\}$, then the relation $R$ and $A$ is
Question 30 :
Which of the following is not an equivalence relation on $Z$?
Question 31 :
If $A=\left\{ 1,2,3 \right\} $, then a relation $R=\left\{ \left( 2,3 \right) \right\} $ on $A$ is
Question 32 :
Let $R$ be the relation over the set of all straight lines in a plane such that ${l}_{1}$ $R$ ${l}_{2}\Leftrightarrow {l}_{1}\bot {l}_{2}$. Then, $R$ is
Question 34 :
$\displaystyle x^{2} = xy$ is a relation (defined on set R) which is<br/> <br/>
Question 35 :
Let $T$ be the set of all triangles in the Euclidean plane, and let a relation $R$ on $T$ be defined as $aRb$, if $a$ is congruent to $b$ for all $a,b\in T$. Then, $R$ is
Question 36 :
If $A=\left\{ a,b,c \right\} $, then the relation $R=\left\{ \left( b,c \right) \right\} $ on $A$ is
Question 37 :
If $R$ is the largest equivalence relation on a set $A$ and $S$ is any relation on $A$, then
Question 38 :
Let $A=\left\{ 1,2,3 \right\} $. Then, the number of equivalence relations containing $(1,2)$ over set A is
Question 39 :
Assertion: Domain of $f(x)$ is singleton.
Reason: Range of $f(x)$ is singleton.
Question 40 :
On the set $N$ of all natural numbers define the relation $R$ by $a R b$ if and only if the G.C.D. of $a$ and $b$ is $2$. Then $R$ is:
Question 41 :
Give a relation R={(1,2), (2,3)} on the set of natural numbers, add a minimum number of ordered pairs.  
Question 42 :
Let $R$ be a reflexive relation on a finite set $A$ having $n$ elements, and let there be $m$ ordered pairs in $R,$ then:
Question 43 :
Let N denote the set of all natural numbers. Define two binary relations on N as $R_1=\{(x, y)\epsilon N\times N : 2x+y=10\}$ and $R_2=\{(x, y)\epsilon N\times N:x+2y=10\}$. Then.
Question 44 :
The minimum number of elements that must be added to the relation $R=\{(1,2,),(2,3)\} $ on the set of natural numbers so that it is an equivalence is
Question 46 :
Let $R = \{(1, 3), (4, 2), (2, 4), (2, 3), (3, 1)\}$ be a relation on the set $A = \{1, 2, 3, 4\}$. The relation $R$ is
Question 47 :
Let $R$ be a reflexive on a finite set $A$ having $n$ elements, and let there be $m$ ordered pairs in $R$. Then
Question 50 :
The function $f:(0,\infty )\rightarrow R$ given by $ f(x)=\cfrac { x }{ x+1 } $ is<br>
Question 51 :
Let $R$ be a reflexive relation in a finite set having $n$ elements and let there be $m$ ordered pairs in $R$. Then,
Question 52 :
If functions $f\left ( x \right )$ and $g\left ( x \right )$ are defined on $R\rightarrow R$ such that<br/>$f(x)=x+3, x$ $\in  $ rational<br/>         $ =4x, x$ $\in $ irrational<br/>$g(x)=x+\sqrt{5}$, x$\in $ irrational<br/>      $  =-x, x$ $\in $ rational<br/>then $\left ( f-g \right )\left ( x \right )$ is<br/>
Question 54 :
Let $f:Z\rightarrow Z$ be given by $f(x)=\begin{cases} \cfrac { x }{ 2 } ,\quad \text{if}\ x \ \text{is even} \\ 0,\quad \text{if }\ x \ \text{is odd} \end{cases}$. Then, $f$ is
Question 55 :
Let $X$ be the set of all persons living in a city. Persons $x, y$ in $X$ are said to be related as $x < y$ if $y$ is at least $5$ years older than $x$. Which one of the following is correct?
Question 57 :
Number of one-one functions from A to B where $n(A)=4, n(B)=5$.
Question 58 :
Let $Z$ be the set of integers and$f:Z\rightarrow Z$ is a bijective function then
Question 59 :
Assertion: $ \displaystyle f:R \rightarrow \left [0,\frac {\pi}{2} \right )$ defined by $ \displaystyle f(x)=\tan^{-1}(x^{2}+x+a)$ is onto for all $ a \in \left ( -\infty ,\dfrac{1}{4} \right )$
Reason: For onto function codomain of $f=$ Range of $f$.
Question 60 :
If A ={1, 3, 5, 7} and B = {1, 2, 3, 4, 5, 6, 7, 8}, then the number of one-to-one functions from A into B is
Question 61 :
$M$ is the set of all $2\times2$ real matrices.$f:M\rightarrow R$is defined by $f(A)=det A$ for all $A$ in $M$ then $f$ is
Question 62 :
The relation 'is a sister of' in the set of human beings is____
Question 63 :
If $f:R\rightarrow R$ given by $f(x)={ x }^{ 3 }+({ a+2)x }^{ 2 }+3ax+5$ is one-one, then $a$ belongs to the interval<br>
Question 64 :
Assertion: Let $f\, :\, R\, \rightarrow\, R$, $f(x)\, =\, x^{3}\, +\, x^{2}\, +\, 100x\, +\, 5\sin x$, then $f(x)$ is bijective.
Reason: $3x^{2}\, +\, 2x\, +\, 95\, >\, 0\, \, x\, \in\, R$.
Question 65 :
Let $A=\left\{ x\in R:-1\le x\le 1 \right\} =B$, then the mapping $f:A\rightarrow B$ given by $f(x)=x\left| x \right| $ is
Question 66 :
The Set $A$ has $4$ elements and the Set $B$ has $5$ elements then the number of injective mappings that can be defined from $A$ to $B$ is
Question 67 :
Let N denote the set of all natural numbers and R a relation on $N\times N$. Which of the following is an equivalence relation?
Question 68 :
The function $f:A\rightarrow B$ given by $f(x),x\in A$, is one to one but not onto. Then;
Question 69 :
The function $f:[0,\infty )\rightarrow R$ given by $f(x)=\cfrac { x }{ x+1 } $ is
Question 70 :
In order that a relation $R$ defined in a non-empty set $A$ is an equivalence relation, it is sufficient that $R$
Question 71 :
Let $f:R\rightarrow R$ be a function defined by $f(x)=\cfrac { { e }^{ \left| x \right|  }-{ e }^{ -x } }{ { e }^{ x }+{ e }^{ -x } } $, then
Question 74 :
Let $R$ a relation on the set $N$ be defined by $\left\{ \left( x,y \right) |x,y\in N,2x+y=41 \right\}$. Then $R$ is
Question 75 :
If $n \geq 2$ then the number of surjections that can be defined from $\{1, 2, 3, .......  n\}$ onto $\{1, 2\}$ is<br/>
Question 76 :
If $f:A\rightarrow B$ given by ${ 3 }^{ f(x) }+{ 2 }^{ -x }=4$ is a bijection, then
Question 77 :
Let f: $X\rightarrow Y$ be a function defined by $f(x)=a \sin \left (x+\dfrac {\pi}{4}\right )+b \cos x+c$. If f is both one-one and onto, then find the sets $X$ and $Y$
Question 78 :
Let $f:\{x, y , z\} \rightarrow \{1, 2, 3\}$ be a one-one mapping such that only one of the following three statements and remaining two are false : $f(x) \neq 2, f(y) =2, f(z) \neq 1$, then
