Relations and Functions

Class 11 NCERT

NCERT

1   If the set $A$ has $3$ elements and the set $B=\{ 3, 4,5\} ,$ then find the number of elements in $( A * B ) ?$

Solution :

It is given that set A has 3 elements and the elements of set $B$ are $3, 4,$ and $5.$$\\$ $\implies $ Number of elements in set $B = 3$$\\$ Number of elements in $( A * B )$$\\$ = (Number of elements in A) * (Number of elements in B) $= 3 * 3 = 9$$\\$ Thus, the number of elements in (A * B } in $9.$

2   If $G = \{7, 8\}$ and $H = \{5, 4, 2\},$ find $G * H$ and $H * G$ .

Solution :

$G = \{7, 8\}$ and $H = \{5, 4, 2\},$$\\$ We know that the Cartesian product $P * Q$ of two non-empty sets $P$ and $Q$ is defined as$\\$ $P * Q -\{ (p , q) : p \in P , q \in Q \}\\ \therefore G*H=\{(7,5),(7,4),(8,5),(8,4),(8,2)\}\\ H*G=\{(5,7),(5,8),(4,7),(4,8),(2,7),(2,8)\}$$\\$

3   State whether each of the following statement are true or false. If the statement is false, rewrite the given statement correctly.$\\$ (i) If P = {m, n} and Q = {n, m}, then P *Q =$\{$( m , n )( n , m )$\}$ .$\\$ (ii) If A and B are non-empty sets, then A *B is a non-empty set of ordered pairs (x, y) such that x $\in $ A and y $\in $ B .$\\$ (iii) If A =$\{ 1, 2 \}$ , B $\{3, 4 \}$ , then A *$\{$ B $\cap \oslash \}=\oslash$

Solution :

4   If $ A=\{-1,1\},$ find $A*A*A.$

Solution :

If is known that for any non-empty set $A,A*A*A$ is defined as$\\$ $A*A*A=\{(a,b,c):a,b,c\in A\}$ It is given that $A=\{-1,1\}$$\\$ $\therefore A*A*A=\{(-1,-1,-1),(-1,-1,1),(-1,1,-1),(-1,1,1),\\ (1,-1,-1),(1,-1,1),(1,1,-1),(1,1,1)\}$

5   If $A*B=\{(a,x),(a,y),(b,x),(b,y)\}$ . Find A and B.

Solution :

If is given that $A*B=\{(a,x),(a,y),(b,x),(b,y)\}$ $\\$ We know that the Cartesian product of two non-empty sets P and Q is defined as$\\$ $P*Q=\{(p,q):p\in P,q\in Q\}$$\\$ $\therefore$ A is the set of all first elements and B is the set of all second elements.$\\$ Thus,$A=\{a,b\}$ and $B=\{x,y\}$

6   If$(\dfrac{x}{3}+1,y-\dfrac{2}{3})=(\dfrac{5}{3},\dfrac{1}{3}),$ find the values of x and y.

Solution :

It is given that $(\dfrac{x}{3}+1,y-\dfrac{2}{3})=(\dfrac{5}{3},\dfrac{1}{3}),$$\\$ Since the ordered pairs are equal, the corresponding elements will also be equal.$\\$ Therefore, $\dfrac{x}{3}+1=\dfrac{5}{3}$ and $ y-\dfrac{2}{3}=\dfrac{1}{3}$$\\$ $\dfrac{x}{3}+1=\dfrac{5}{3}\\ \implies \dfrac{x}{3}=\dfrac{5}{3}-1 y-\dfrac{2}{3}=\dfrac{1}{3}\\ \implies \dfrac{x}{3}=\dfrac{2}{3} \implies y=\dfrac{1}{3}+\dfrac{2}{3}\\ \implies x=2 \implies y=1\\ \therefore x=2$ and $ y=1 $