Question

If A = (a, b, c, d), B = (p, q, r, s) then which of the following are relation from a to b
$a){{R}_{1}}=\left\{ \left( a,p \right),\left( b,r \right),\left( c,s \right) \right\}$
$b){{R}_{2}}=\left\{ \left( q,b \right),\left( c,s \right),\left( d,r \right) \right\}$
$c){{R}_{3}}=\left\{ \left( a,p \right),\left( a,q \right),\left( d,p \right),\left( c,r \right),\left( b,r \right) \right\}$
$d){{R}_{4}}=\left\{ \left( a,p \right),\left( a,q \right),\left( b,s \right),\left( s,b \right) \right\}$

Answer 1

Hint: A relation R is called a relation from A to B if the elements of $R=\{(x,y):x\in A,y\in B\}$ . Hence for each option we will check if the condition is satisfied and accordingly answer if it is a relation from A to B.

Complete step-by-step answer:
Now consider option $a){{R}_{1}}=\left\{ \left( a,p \right),\left( b,r \right),\left( c,s \right) \right\}$
Here the relation is ${{R}_{1}}$ now let us consider all three elements
In $\left( a,p \right)$ we have $a\in A,p\in B$
In $\left( b,r \right)$ we have $b\in A,p\in B$
In $\left( c,s \right)$ we have $c\in A,q\in B$
Hence all the elements are of the form $(x,y):x\in A,y\in B$ hence we have ${{R}_{1}}$ is a relation
Now consider option $b){{R}_{2}}=\left\{ \left( q,b \right),\left( c,s \right),\left( d,r \right) \right\}$
Here if we consider the first element itself $(q,b)$ is such that \[q\in B,b\in A\] this element is not in the form of $(x,y):x\in A,y\in B$ . hence ${{R}_{2}}$ is not a relation from A to B
Now consider option $c){{R}_{3}}=\left\{ \left( a,p \right),\left( a,q \right),\left( d,p \right),\left( c,r \right),\left( b,r \right) \right\}$
Here also we have the elements $\left( a,p \right),\left( a,q \right),\left( d,p \right),\left( c,r \right),\left( b,r \right)$.
Now $a,d,c,b$ all belong to set A similarly $p,q,r$ all belong to set B
Hence all the elements in ${{R}_{3}}$ are of the form $(x,y):x\in A,y\in B$
Hence we have ${{R}_{3}}$ is a relation from A to B
Now consider option $d){{R}_{4}}=\left\{ \left( a,p \right),\left( a,q \right),\left( b,s \right),\left( s,b \right) \right\}$
Now here if we consider the last element $(s,b)$
Here \[s\in B\] and \[b\in A\]
Hence the element is not in the form of $(x,y):x\in A,y\in B$
Hence ${{R}_{4}}$ is not a relation from A to B.

Note: Now while checking if it is a relation from A to B we have to check that if all the elements are in the form of $(x,y):x\in A,y\in B$. Even if there is one element which does not follow the condition then we can say that the relation is not a relation from A to B.