Definition of Relations

Q. Let $X$ be a set with exactly $5$ elements and $Y$ be a set with exactly $7$ elements. If $\alpha$ is the number of one-one functions from $X$ to $Y$ and $\beta$ is the number of onto functions from $Y$ to $X,$ then the value of $\frac{1}{5!}(\beta -\alpha )$ is

Q. If $A=\{x:x^2-5x+6=0\}$ and $B=\{y:y\in Z,3<|y-2|\le 5\},$ then the number of relations from $A$ to $B$ is

Q. If a function defined from $A$ to $B$ as follows


Then the correct options is/are

1. Domain of function is $\{1,2,3,4,5\}$
2. Co-domain of function is $\{2,4,6,8,10\}$
3. Range of function is $\{2,4,6,8,10\}$
4. Co domain and range of the given function is same.

Q. If a function is defined from $A$ to $B$ as


then the total number of elements in domain of function is

Q. Which of the following is an identity relation on the set $A=\{a,b,c\}$?

1. $R=\left\{\right(a,a),(b,b\left)\right\}$
2. $R=\left\{\right(a,b),(b,c\left)\right\}$
3. $R=\left\{\right(a,b),(b,c),(c,a\left)\right\}$
4. $R=\left\{\right(a,a),(b,b),(c,c\left)\right\}$

Q. Let $A=\{1,3,5,7\}$ and $B=\{2,4,6,8\}$ be two sets and $R$ be a relation from $A$ to $B$ defined by the phrase ${}^{\prime \prime }(x,y)\in R:x<y^{\prime \prime },$ then the number of elements in the range of $R$ is

Q. A relation $R$ defined on the set of non-zero complex numbers as $z_{1}R{z}_2\text{iff}\frac{z_1-z_2}{z_1+z_2}$ is real, then $R$ is

1. a reflexive relation
2. an equivalence relation
3. a symmetric relation
4. a transitive relation

Q. Let $N$ denote the set of natural numbers and $R$ be a relation on $N\times N$ defined by
$(a,b)R(c,d)⟺ad(b+c)=bc(a+d).$ Then on $N\times N,R$ is

1. An equivalence relation
2. Reflexive and symmetric relation only
3. Symmetric and transitive relation only
4. Transitive relation only

Q. If $A=\{1,2,3,4,5,6\},B=\{3,6,9,12\},C=\{6,12,18,20\}$, then $n\left\{\right(A\times B)\cap (A\times C\left)\right\}=$

1. $12$
2. $24$
3. $48$
4. $36$

Q. Let $R=\left(\left(x,y\right):x,y\in Z,y=2x-4\right\}.$ If (p, -2) and $\left(q^2,4\right)\in R$ and pq < 0 , then the value of p =
and q =

Q. Which among the following relations on $Z$ is an equivalence relation

1. $xRy\iff |x|=|y|$
2. $xRy\iff x>y$
3. $xRy\iff x<y$
4. $xRy\iff x\ge y$

Q. Let $R=\left(\left(x,y\right):x,y\in Z,y=2x-4\right\}.$ If (p, -2) and $\left(q^2,4\right)\in R$ and pq < 0 , then the value of p =
and q =

Q. If $|x^2+3x-4|=|x+4|$, then $x=$

1. $-4$
2. $2$
3. $-2$
4. $0$

Q. If a set $A$ is a subset of another set $B,$ then , which means

1. $A\subseteq B$
2. $B\supseteq A$
3. $\mathrm{if}a\in A\Rightarrow a\in B$
4. $\mathrm{if}a\in B\Rightarrow a\in A$

Q.

If relation R is defined as "Is of the same color" on set of objects. Then the total number of equivalence classes on the set with respect to R is equal to.

1. Number of distinct objects in the set.

2. (Number of objects)${}^2$

3. Number of distinct colors in the set

4. (Number of distinct colors)${}^2$

Q. Let $n\left(A\right)=m$ and $n\left(B\right)=n.$ Then, the total number of non-empty relations that can be defined from $A$ to $B$ is .

1. $m^n$
2. $mn-1$
3. $2^{mn}-1$
4. $n^m-1$
5. $2^{mn}$

Q.

If $A=\{5,7,9,11\},B=\{9,10\}$ and $aRb$ means $a<b$ where $a\in A,b\in B,$ then which of the following are true?

1. Co-domain of $R$ is $\{9,10\}.$

2. Range of $R=$Co-domain of $R$

3. $R=\left\{\right(5,9),(5,10),(7,9),(7,10),(9,10\left)\right\}$

4. Domain of $R$ is $\{5,7,9\}.$

Q. If A and B are two sets having 3 elements in common. If n(A)=6 and n(B)=4, then $n\left(\left(A\times B\right)\cap \left(B\times A\right)\right]=$

Q.

Let X = {1, 2, 3, 4, 5} and Y = {1, 3, 5, 7, 9}. Which of the following is/are relations from X to Y

1. $R_1=\left\{\right(x,y)y=2+x,x\in X,y\in Y\}$

2. $R_2=\left\{\right(1,1),(2,1),(3,3),(4,3),(5,5\left)\right\}$

3. $R_3=\left\{\right(1,1),(1,3),(3,5),(3,7),(5,7\left)\right\}$

4. $R_4=\left\{\right(1,3),(2,5),(2,4),(7,9\left)\right\}$

Q. If $A=\{x:x^2-5x+6=0\}$ and $B=\{y:y\in Z,3<|y-2|\le 5\},$ then the number of relations from $A$ to $B$ is

Q. $\text{For any two sets}A\&B\text{if}$
$A\subset B,\text{then}A\cap B=$

1. $A$
2. $B$
3. $A\cup B$