Question

If {\text{A = }}\left\\{ {{\text{a,b,c,d}}} \right\\}, {\text{B = }}\left\\{ {{\text{p,q,r,s}}} \right\\}then which of the following are relations from A to B?
A) R1 = \left\\{ {\left( {a,p} \right),\left( {b,q} \right),\left( {c,s} \right)} \right\\}
B) R2 = \left\\{ {\left( {q,b} \right),\left( {c,s} \right),\left( {d,r} \right)} \right\\}
C) R3 = \left\\{ {\left( {a,p} \right),\left( {a,q} \right),\left( {d,p} \right),\left( {c,r} \right),\left( {d,r} \right)} \right\\}
D) R4 = \left\\{ {\left( {a,p} \right),\left( {q,a} \right),\left( {b,s} \right),\left( {s,b} \right)} \right\\}

Answer 1

Here we have to let a relation R from A to B such that ${\text{R}} \subseteq {\text{A}} \times {\text{B}}$ and the domain of
${\text{R = \\{ a:a}} \in {\text{A (a,b)}} \in {{R\;for some\;b}} \in {\text{B\\} }}$and the range of ${\text{R = \\{ \;b:b}} \in {\text{B,(a,b)}} \in {{R\;for some\;a}} \in {\text{A\\} }}$
And then calculate ${\text{A}} \times {\text{B}}$

Complete step by step solution:
We are given two sets {\text{A = }}\left\\{ {{\text{a,b,c,d}}} \right\\} and {\text{B = }}\left\\{ {{\text{p,q,r,s}}} \right\\} and we have to make a relation from A to B.
Let the relation from A to B be R such that ${\text{R}} \subseteq {\text{A}} \times {\text{B}}$ and the domain of
${\text{R = \\{ a:a}} \in {\text{A (a,b)}} \in {{R\;for some\;b}} \in {\text{B\\} }}$and the range of ${\text{R = \\{ \;b:b}} \in {\text{B,(a,b)}} \in {{R\;for some\;a}} \in {\text{A\\} }}$
Then ,
First calculating ${\text{A}} \times {\text{B}}$ we get:
{\text{A}} \times {\text{B}} = \left\\{ {\left( {{\text{a,p}}} \right){\text{,}}\left( {a,q} \right),\left( {a,r} \right),\left( {a,s} \right),\left( {b,p} \right),\left( {{\text{b,q}}} \right){\text{,}}\left( {b,r} \right){\text{,}}\left( {b,s} \right){\text{,}}\left( {c,p} \right){\text{,}}\left( {c,q} \right){\text{,}}\left( {{\text{c,r}}} \right){\text{,}}\left( {c,s} \right){\text{,}}\left( {d,p} \right){\text{,}}\left( {d,q} \right){\text{,}}\left( {d,r} \right){\text{,}}\left( {{\text{d,s}}} \right)} \right\\}
Since R should be subset of ${\text{A}} \times {\text{B}}$
${\text{R}} \subseteq {\text{A}} \times {\text{B}}$
Therefore checking for option A we get:-
R1 = \left\\{ {\left( {a,p} \right),\left( {b,q} \right),\left( {c,s} \right)} \right\\}
Since,
${\text{R1}} \subseteq {\text{A}} \times {\text{B}}$
Therefore, option A is correct.
Now checking for option B we get:-
R2 = \left\\{ {\left( {q,b} \right),\left( {c,s} \right),\left( {d,r} \right)} \right\\}
Since,
${\text{R2}} \not\subset {\text{A}} \times {\text{B}}$
Therefore option B is incorrect.
Checking for option C we get:-
R3 = \left\\{ {\left( {a,p} \right),\left( {a,q} \right),\left( {d,p} \right),\left( {c,r} \right),\left( {d,r} \right)} \right\\}
Since,
${\text{R3}} \subseteq {\text{A}} \times {\text{B}}$
Therefore it is a correct option.
Now checking for option D we get:-
R4 = \left\\{ {\left( {a,p} \right),\left( {q,a} \right),\left( {b,s} \right),\left( {s,b} \right)} \right\\}
Since,
${\text{R4}} \not\subset {\text{A}} \times {\text{B}}$
Therefore it is incorrect option.

Hence, option A and option C are correct options.

Note:
The relation R should be a subset of ${\text{A}} \times {\text{B}}$ to be a relation from A to B in which the first element of the ordered pair should be from set A and the second element should be from set B.