KC Sinha Mathematics Solution Class 11 Chapter 2 Relations Exercise 2.1

 Exercise 2.1 

Page no- 2.13

Type 1 

Question 1

(a)If $A=\{a, b, c\}$ and $B=\{x, y, z\}$, then find  (i) $A \times B$(ii) $B \times A$(iii) $A \times A$

(b) If $A=\{0\}, B=\{1,\{2\}\}$, then find $A \times B$.

(c) Find $A \times A$ when $A=\{1,2,3\}$.

(d) If $A=\{1,2,3,5\}$ and $B=\{0,1,3,5\}$, then find $A \times B$ and $B \times A$.

(c) If $A=\{a, b, c\}$ and $B=\{r\}$, form the sets $A \times B$ and $B \times A$.

Are these two products equal ?

(f) If $G=\{7,8]$ and $H=\{5,4,2\}$, find $G \times H$ and $H \times G$.

Page no- 2.13

Question 2

If $A=\{1,2,3\}, B=\{1,2\}, C=\{3,5,7\}$, then find :

(i) $A \times(B \cup C)$

(ii) $A \times(B \cap C)$

(iii) $(A \cup B) \times C$

(iv) $(A \cap B) \times C$

(v) $(A \times B) \cup(A \times C)$

(vi) $(A \times B) \cap(A \times C)$

Question 3

If $A=\{a, b\}, B=(2,3\}, C=\{3,4\}$, then find

(i) $A \times(B \cup C)$

(ii) $A \times(B \cap C)$

(iii) $(A \times B) \cap(A \times C)$

Question 4

If $A=\{4,5\}, B=\{6,7\}, C=\{7,8\}$, then find $(A \times B) \cup(B \times C)$

Question 5

If $A=|1,2|$ and $B=\{1,3\}$, find $(A \times B) \cup(B \times A)$

Question 6

If $A=\{1,2,3\}, B=\{3,4\}, C=\{4,5,6\}$, then find

(i) $(A \times B) \cup(B \times C)$

(ii) $(A \times B) \cap(B \times C)$

(iii) $(A \times B) \cup(A \times C)$

(iv) $(A \times B) \cap(A \times C)$

(v) $A \times(B \cap C)$

(vi) $A \times(B \cup C)$

Question 7

If $A=\{1,2,3, a, b\}, B=\{2,3, a, c, d\}, C=\{1, b, a, d, e\}$. then test the validity of $(A-B) \times C=(A \times C)-(B \times C)$

Question 8

If $A=\{2,3,4\}, B=\{3,5\}, C=\{2,6\}$. Then verify that

(i) $A \times(B \cup C)=(A \times B) \cup(A \times C)$

(ii) $A \times(B \cap C)=(A \times B) \cap(A \times C)$

(iii) $C \times(A-B)=C \times A-C \times B$

Question 9

Let $A=\{1,2\}, B=\{1,2,3,4\}, C=\{5,6\}$ and $D=\{5,6,7,8\}$. Verify that

(i) $A \times C \subseteq B \times D$

(ii) $A \times(B \cap C)=(A \times B) \cap(A \times C)$

Question 10

If $a \in\{2,4,6,9\}$ and $b \in\{4,6,18,27\}$. then form the set of all ordered pairs $(a, b)$ such that $a$ divides $b$ and $a<b$.

Question 11

If $a \in\{-1,2,3,4,5\}$ and $b \in\{0,3,6\}$, write down the set of all ordered pairs $(a, b)$ such that $a+b=5$

Question 12

If the ordered pairs $(x,-1)$ and $(5, y)$ belong to the set $\{(a, b): b=2 a-3\}$, find the values of $x$ and $y$.

Question 13

Express the following sets, Explicitly

(i) $\left\{(x, y): x^{2}+y^{2}=25, x, y \in N\right\}$

(ii) $\{(x, y): 2 x+3 y=15, x, y \in W\}$

Question 14

State whether following statements are true or false. If it is false, rewrite it correctly.

(i) If $A=\{1,2\}, B=\{3,4\}$, then $A \times(B \cup \varphi)=\varphi$

(ii) If $A=\{1,2\}, B=\{3,4\}$, then $A \times(B \cap \varphi)=\varphi$

(iii) If $A=\{2,3\}$ and $B=\{4,5\}$, then $A \times B=\{(2,4),(3,5)\}$

Question 15

If $A=\{-1,1\}$, find $A \times A \times A$

Question 16

If $A \times B=\{(p, q),(p, r),(m, q),(m, r)\}$, find $A$ and $B$.

Page no- 2.14

Type 2

Question 17

(i) If $(x+2,4)=(5,2 x+y)$. find $x$ and $y$.

(ii) If $(x-2,2 y+1)=(y-1, x+2)$, find $x$ and $y$.

(iii) If $(x+1, y-2)=(3,1)$, find $x$ and $y$.

(iv) If $\left(\frac{a}{3}, b+5\right)=(-1,-2)$, find $a$ and $b$.

Question 18

If $A=\{a, b, c\}$ and $B=\{p, q\}$, then find

(i) $n(A \times B)$

(ii) $n(B \times A)$

(iii) $n(A \times A)$

Question 19

If $A=\{a, b, c, d\}$ and $B$ is equivalent to $A$, then find the number of elements in

(i) $A \times B$

(ii) $B \times B$

Question 20

Let $A$ and $B$ be two sets such that $n(A)=3$ and $n(B)=2$. If $(x, 1),(y, 2),(z, 1)$ are in $A \times B$, find $A$ and $B$, where $x, y, z$ are distinct elements.

Question 21

If $A=\{a, b, c\}$ and some elements, of $A \times B$ are $(a, p),(b, q),(c, p)$. Write down the remaining elements of $A \times B$ if $n(A \times B)=6$.

Question 22

If $B=\{2,3,5\}$ and $(a, 2),(b, 3),(c, 5)$ are in $A \times B$, find $A$ and the remaining elements of $A \times B$ such that $n(A \times B)$ is least.

Type 3

Question 23

If $A=\{1,2,4\}$ and $B=\{1,2,3\}$, represent graphically the following sets

(i) $A \times B$

(ii) $A \times A$

Question 24

If $A=\{1,2,3\}, B=\{4,5\}$, represent the following products by arrow diagrams :

(i) $A \times B$

(ii) $B \times B$

Type 4

Question 25

Prove that (i) $(A \cup B) \times C=(A \times C) \cup(B \times C)$

(ii) $(A \cap B) \times C=(A \times C) \cap(B \times C)$

Question 26

If $A \times B \subseteq X \times Y$ and $A \times B \neq \phi$, then prove that $A \subseteq X$ and $B \subseteq Y$.

Question 27

Prove that $A \times A=B \times B \Rightarrow A=B$