Definition of Relations

Q.

If A and B be symmetric matrices of the same order, then AB-BA will be

1. Null matrix

2. None of these

3. Symmetric matrix

4. Skew symmetric matrix

Q. Let $A=\{1,2,3,4,5,6\}.$ Define a relation $R$ from $A$ to $A$ by $R=\left\{\right(x,y):y=x+1\}$

$\left(i\right)$ Depict this relation using an arrow diagram
$\left(ii\right)$ Write down the domain, codomain, and range of $R$.

Q.

If $P\left(A\right)$denotes the power set of $A$and $A$ is the void set, then what is the number of elements in $P\left\{P\right\{P\{P\left(A\right)\left\}\right\}\}$?

1. $0$

2. $1$

3. $4$

4. $16$

Q.

Prove that on the set of integers, the relation R defined as aRb if and only if a=±b is an equivalence relation

Q.

Let R be a relation from a set A to a set B. then

1. $R=A\cap B$

2. $R=A\cup B$

3. $R\subseteq A\times B$

4. $R\subseteq B\times A$

Q. The domain of the function $f\left(x\right)=\sqrt{x^{12}-x^3+x^4-x+1}$ is

1. $(1,\infty )$
2. $(-\infty ,0)$
3. $(-\infty ,0)\cup (1,\infty )$
4. $R$

Q.

A = {1, 2, 3, 5} and B = {4, 6, 9}. Define a relation R from A to B by R = {(x, y) : the difference between x and y is odd, $xϵA,yϵB$}.

Write R in Roster form.

Q.

A relation $R$ is defined from $\left\{2,3,4,5\right\}$ to $\left\{3,6,7,10\right\}$ by $xRy\Rightarrow x$ is relatively prime to $y$. Then the domain of $R$ is

1. $\left\{2,3,5\right\}$

2. $\left\{3,5\right\}$

3. $\left\{2,3,4\right\}$

4. $\left\{2,3,4,5\right\}$

Q. Let N be the set of natural numbers and the relation R be defined on N such that
R={(x, y);y=2x, x, y $ϵ$ N}.
What is the domain, codomain, and range of R?
Is this relation a function?

Q. Let $R$ be the relation “is congruent to” on set of all triangles in a plane. Then $R$ is

[1 mark]

1. reflexive and symmetric only
2. symmetric only
3. transitive only
4. an equivalence relation

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.

Write the following relations as the sets of ordered pairs :

(i) A relation R from the set {2, 3, 4, 5, 6} to the set {1, 2, 3} defined by x = 2y.

(ii) A relation R from the set {1, 2, 3, 4, 5, 6, 7} defined by $(x,y)ϵR\iff x$ is relatively prime to y.

(iii) A relation R on the set {0, 1, 2, ...., 10} defined by 2x + 3y = 12.

(iv) A relation R from a set A = {5, 6, 7, 8} to the set B = {10, 12, 15, 16, 18} defined by $(x,y)ϵR\iff x$ divides y.

Q. If the area of the bounded region
$R=\left\{\right(x,y):max\{0,{\log }_{e}x\}\le y\le 2^x,\frac{1}{2}\le x\le 2\}$ is, $\alpha ({\log }_{e}2{)}^{-1}+\beta ({\log }_{e}2)+\gamma ,$ then the value of $(\alpha +\beta -2\gamma {)}^2$ is equal to

1. $2$
2. $1$
3. $8$
4. $4$

Q.

Let A = {a, b}. List all relations on A and find their number.

Q. $A=\{5,7,9\},$ then which of the following is the identity relation on $A$?

1. $R_2=\left\{\right(9,7),(7,5),(5,9),(5,5),(7,7),(9,9\left)\right\}$
2. $R_3=\left\{\right(5,5),(7,7),(9,9\left)\right\}$
3. $R_4=\left\{\right(5,7),(7,9),(9,5\left)\right\}$
4. $R_1=\left\{\right(5,5),(7,7),(9,9),(5,7),(7,9),(9,5\left)\right\}$

Q.

Let $f:R-\{-\frac{4}{3}\}\to R$ be a function defined as $f\left(x\right)=\frac{4x}{3x+4},x\ne -\frac{4}{3}$. The inverse of f is the map g: Range $f\to R-\{-\frac{4}{3}\}$ is given by
$\left(a\right)g\left(y\right)=\frac{3y}{3-4y}\left(b\right)g\left(y\right)=\frac{4y}{4-3y}\left(c\right)g\left(y\right)=\frac{4y}{3-4y}\left(d\right)g\left(y\right)=\frac{3y}{4-3y}$

Q. Let $N=10800$

$\text{I}.$ Total number of divisors of $N$ is $60$
$\text{II}.$ The number of even divisors of $N$ is $24$
$\text{III}.$ The number of divisors of $N$ which are multiples of $15$ is $30$

Then which of the following statements are true?

1. only $\text{I}$
2. only $\text{I and}\text{III}$
3. only $\text{I and}\text{II}$
4. All of the above

Q.

How many different $10$-letter words (real or imaginary) can be formed from the following letters? $\text{O}$, $\text{Z}$, $\text{Q}$, $\text{Y}$, $\text{N}$, $\text{U}$, $\text{Z}$, $\text{J}$, $\text{W}$, $\text{V}$ ten-letter words (real or imaginary) can be formed with the given letters. (Type a whole number.)

Q.

A relation n$\varphi$ from C to R is defined by $x\varphi y\iff |x|=y.$ Which one is correct ?

1. $(2+3i)\varphi 13$

2. $3\varphi (-3)$

3. $(1+i)\varphi 2$

4. $i\varphi 1.$

Q.

Find the inverse relation $R^{-1}$ in each of the following cases :

(i) R = {(1, 2), (1, 3), (2, 3), (3, 2), (5, 6)}

(ii) $R=\left\{\right(x,y):x,yϵN,x+2y=8\}$

(iii) R is a relation form {11, 12, 13} to {8, 10, 12} defined by y = x - 3.

Q.

Let A = {3, 5} and B = {7, 11}. Let $R=\left\{\right(a,b):aϵA,bϵB,a-b$ is odd}. Show that R is an empty relation from A into B.

Q.

Define a relation R on the set N of natural numbers by R = {(x, y) : y = x + 5, x} is a natural number less than $4,x,yϵN$}

Depict this relationship using roster form

Q. Let $A$ and $B$ be any two $3\times 3$ matrices. If $A$ is symmetric and $B$ is skew-symmetric, then the matrix $AB-BA$ is always :

1. Symmetric
2. Skew symmetric
3. Lower triangular matrix
4. Neither symmetric nor skew symmetric

Q. Let $A=\{1,4,9,25\}$ and $B=\{-5,-3,-2,-1,1,2,3,5\}$, if relation from $A$ to $B$ is $R=\left\{\right(1,1),(1,-1),(4,2),(4,-2),(9,3),(9,-3),(25,5),(25,-5\left)\right\}$, then the set builder form of relation is

1. $R=\left\{\right(x,y);y=x^2,x\in A,y\in B\}$
2. $R=\left\{\right(x,y);y=|x|,x\in A,y\in B\}$
3. $R=\left\{\right(x,y);y^2=x,x\in A,y\in B\}$
4. $R=\left\{\right(x,y);y=\sqrt{x},x\in A,y\in B\}$

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 $f\left(x\right)=\frac{x^4-\lambda x^3-3x^2+3x\lambda }{x-\lambda },x\in R-\left\{\lambda \right\}$. If the range of $f\left(x\right)$ is $R$, then the complete set of values of $\lambda$ is
​​​​​​​(correct answer + 1, wrong answer - 0.25)

1. $[0,4]$
2. $(1,3)$
3. $(-2,2)$
4. $[-2,2]$

Q.

If |z + 2i| $\le 1$ and $z_1$ = 6-3i then the maximum value of |iz + $z_1$ - 4 | is equal to

1. 2

2. 6

3. 3

4. 

Q.

For the relation $R_1$ defined on R by the rule $(a,b)ϵR_1\iff 1+ab>0.$

Prove that : $(a,b)ϵR_1$ and $(b,c)ϵR_1$

$\Rightarrow (a,c)ϵR_1$ is not true for all $a,bcϵR$

Q.

Let A and B be two sets show that the set A×B and B×A an element in common IFF the sets A and B have an element in common.

Q.

Let $A=\left\{1,2,3\right\}$ $B=\left\{1,3,5\right\}$. If relational $R$ from $A$ to $B$ is given by $R=\left\{\left(1,3\right),\left(2,5\right),(3,3)\right\}$. Then $R^{-1}$ is

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

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

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

4. None of these