# Definition of Relations

## Trending Questions

**Q.**

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

Null matrix

None of these

Symmetric matrix

Skew symmetric matrix

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

(i) Depict this relation using an arrow diagram

(ii) 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 $P\left\{P\right\{P\{P\left(A\right)\left\}\right\}\}$?

$0$

$1$

$4$

$16$

**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

R=A∩B

R=A∪B

R⊆A×B

R⊆B×A

**Q.**The domain of the function f(x)=√x12−x3+x4−x+1 is

- (1, ∞)
- (−∞, 0)
- (−∞, 0) ∪(1, ∞)
- 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

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

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

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

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

**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]

- reflexive and symmetric only
- symmetric only
- transitive only
- 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((A×B)∩(B×A)]=

**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⇔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⇔x divides y.

**Q.**If the area of the bounded region

R={(x, y):max{0, logex}≤y≤2x, 12≤x≤2} is, α(loge2)−1+β(loge2)+γ, then the value of (α+β−2γ)2 is equal to

- 2
- 1
- 8
- 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?

- R2={(9, 7), (7, 5), (5, 9), (5, 5), (7, 7), (9, 9)}
- R3={(5, 5), (7, 7), (9, 9)}
- R4={(5, 7), (7, 9), (9, 5)}
- R1={(5, 5), (7, 7), (9, 9), (5, 7), (7, 9), (9, 5)}

**Q.**

Let f:R−{−43}→R be a function defined as f(x)=4x3x+4, x≠−43. The inverse of f is the map g: Range f→R−{−43} is given by

(a)g(y)=3y3−4y(b)g(y)=4y4−3y(c)g(y)=4y3−4y(d)g(y)=3y4−3y

**Q.**Let N=10800

I. Total number of divisors of N is 60

II. The number of even divisors of N is 24

III. The number of divisors of N which are multiples of 15 is 30

Then which of the following statements are true?

- only I
- only I and III
- only I and II
- 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 nϕ from C to R is defined by xϕy⇔|x|=y. Which one is correct ?

(2+3i)ϕ13

3ϕ(−3)

(1+i)ϕ2

iϕ1.

**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={(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={(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 3×3 matrices. If A is symmetric and B is skew-symmetric, then the matrix AB−BA is always :

- Symmetric
- Skew symmetric
- Lower triangular matrix
- 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={(1, 1), (1, −1), (4, 2), (4, −2), (9, 3), (9, −3), (25, 5), (25, −5)}, then the set builder form of relation is

- R={(x, y);y=x2, x∈A, y∈B}
- R={(x, y);y=|x|, x∈A, y∈B}
- R={(x, y);y2=x, x∈A, y∈B}
- R={(x, y);y=√x, x∈A, y∈B}

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

- R={(a, a), (b, b)}
- R={(a, b), (b, c)}
- R={(a, b), (b, c), (c, a)}
- R={(a, a), (b, b), (c, c)}

**Q.**Let f(x)=x4−λx3−3x2+3xλx−λ, x∈R−{λ}. If the range of f(x) is R, then the complete set of values of λ is

(correct answer + 1, wrong answer - 0.25)

- [0, 4]
- (1, 3)
- (−2, 2)
- [−2, 2]

**Q.**

If |z + 2i| ≤1 and z1 = 6-3i then the maximum value of |iz + z1 - 4 | is equal to

2

6

3

**Q.**

For the relation R1 defined on R by the rule (a, b)ϵR1⇔1+ab>0.

Prove that : (a, b)ϵR1 and (b, c)ϵR1

⇒(a, c)ϵR1 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

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

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

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

None of these