Definition of Relations

Q.

A = { 1, 2, 3, 4, 5} and B = {a, b}. The number of relations from A to B is ___ .

1. 128

2. 512

3. 1024

4. 256

Q. If $|z_1|=|z_2|=|z_3|=...=|z_n|=1$ then prove that
$|\frac{1}{z_1+\frac{1}{z_2}+\frac{1}{z_3}}+...+\frac{1}{z_n}|=|z_1+z_2+z_3+...+z_n|.$

Q.

Let A = {1, 2, 3, ....., 14}. Define a relation R from A to A by R = {(x, y): 3x - y = 0, where $x,yϵA$} Write down its domain, codomain and range.

Q.

Which of the following relations are functions? Give reasons. If it is a function determine its domain and range.

(i) {(2, 1), (5, 1), (8, 1), (11, 1), (14, 1), (17, 1)}

(ii) {(2, 1), (4, 2), (6, 3), (8, 4), (10, 5), (12, 6), (14, 7)}

(iii) {(1, 3), (1, 5), (2, 5)}

Q.

Determine the domain and range of the relation R defined by:

$R=\left\{\right(x,x+5):xϵ\{0,1,2,3,4,5\left\}\right\}$

Q.

Given two finite sets A and B such that n(A) = 3, n(B) = 3. Then total number of relations from A to B is _____.

1. 8

2. 4

3. 512

4. 6

Q.

Let N be the set of all natural numbers. Let $R=\left\{\right(a,b):a,bϵN\text{and}2a+b=10\}$. Show that R is a binary relation on N. Find its domain, range and co-domain.

Q.

Let A = {1, 2, 3, 4, 6}. Let R be the relation on A defined by $\left\{\right(a,b):a,bϵA,b$ is exactly divisible by a}.

(i) Write R in roster form.

(ii) Find the domain of R.

(iii) Find the range of R.

Q. Two sets A and B have p and q elements respectively. Then the number of relations from B to A is

1. $q^p$
2. $p+q$
3. $pq$
4. $2^{pq}$

Q.

If A = {2, 3, 5, 6}, B = {4, 8, 15, 17} a R b $\Rightarrow$ a divides b. Find domain and range.

Q. Set A = {5, 9, 11} and set B = {5, 9, 10}. Let aRb be such that a<b where $a\in A,b\in Band(a,b)\in R$.

1. {5, 9, 10}
2. {5, 9, 11}
3. {9, 10}
4. {(5, 9)(5, 10)(9, 10)}

Q.

Let R be the relation on Z defined by R = {(a, b): a, b $\in$ Z a - b is an integer}. Find the domain and range of R.

1. Domain of R = ∅ Range of R = ∅

2. Domain of R = Z, Range of R = Z

3. Domain of R = {1, 2, 3, 4, 5} Range of R = {0, 1, 2, 3, 4, 5}

4. None of these

Q. Which of the following can be the elements of a relation (R) from a set A = {1, 2, 3, 4} to set B = {a, b, c}.

1. (1, a)
2. (b, 2)
3. (2, c)
4. (c, 4)
5. (4, b)
6. (c, 1)
7. (a, 3)
8. (3, b)
9. (1, b)
10. (a, 2)

Q.

If A = {5, 7, 9, 11}, B = {9, 10} let a R b means a < b. a $\in$ A, (a, b) $\in$ R, b $\in$ B. Then

1. Co domain of R = {9, 10}

2. Range of R = {9, 10}

3. Relation of R = {(5, 9), (5, 10), (7, 9), (7, 10), (9, 10)}

4. Domain of R = {5, 7, 9}

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. If R is a relation from set A having four elements to set B having two elements, find the number of relations fron A to B.

Q. If $A=\left\{\left(x,y\right):y=\frac{1}{x},0\ne x\in R\right\}$ and $B=\left\{\left(x,y\right):y=-x,x\in R\right\}$, then write $A\cap B$.

Q.

If $A\cap B^{\prime }=\varphi$ then prove that $A=A\cap B$ and hence show that $A\subseteq B$.

Q.

Find the value of r, if ${}^5{P}_r=2^6{P}_{r-1}$

Q.

The relation R defined on the set of natural numbers as {(a, b) : a differs from b by 3}, is given by

1. {(1, 4, (2, 5), (3, 6), .....}

2. {(4, 1), (5, 2), (6, 3), .....}

3. {(1, 3), (2, 6), (3, 9), ..}

4. None of these

Q.

If A = {2, 3, 5, 6}, B = {4, 8, 15, 17} a R b $\Rightarrow$ a divides b. Find domain and range.

1. Domain = {2, 5, 6}, Range = {4, 15, 17}

2. Domain = {2, 3, 5, 6}, Range = {4, 8, 15, 17}

3. Domain = {2, 3, 5}, Range = {4, 8, 15}

4. Domain = {3, 5, 6}, Range = {8, 15, 17}

Q.

From the sets given below, select equal sets :

A = {2, 4, 8, 12}, B = {1, 2, 3, 4},

C = {4, 8, 12, 14}, D = {3, 1, 4, 2},

E = {-1, 1}, F = {0, a}

G = {1, -1}, H = {0, 1}

Q.

The relation R defined on the set of natural numbers as {(a, b) : a differs from b by 3}, is given by

1. {(1, 4, (2, 5), (3, 6), .....}

2. None of these

3. {(4, 1), (5, 2), (6, 3), .....}

4. {(1, 3), (2, 6), (3, 9), ..}

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 R is a relation defined on the set Z of integers by the rule (x, y) ∈ R ⇔ x² + y² = 9, then write domain of R.

Q.

With reference to a universal set, the inclusion of a subset in another, is relation, which is

1. Symmetric only

2. Reflexive only

3. Equivalence relation

4. None of these

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 are relations from the set $A=\{1,2,3,4\}$ to set $B=\{a,b,c\}$?

1. $\left\{\right(1,a),(1,b),(1,c\left)\right\}$

2. $\left\{\right(2,a),(2,b),(2,c\left)\right\}$

3. $\left\{\right(3,a),(3,b),(3,c\left)\right\}$

4. $\left\{\right(4,a),(4,b),(4,c\left)\right\}$

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

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

Q. Two sets A and B have p and q elements respectively. Then the number of relations from B to A is

1. $p+q$
2. $pq$
3. $q^p$
4. $2^{pq}$