Question

A relation from P to Q is?
(a) A universal set of P $\times$ Q
(b) P $\times$ Q
(c) An equivalent set of P $\times$ Q
(d) A subset of P $\times$ Q

Answer 1

Here first we have to understand the definition of a relation. Further, we will take an example of sets P and Q containing some elements, defined by some relation and try to find the set P $\times$ Q to check the correct option. If all the elements in the relation defined from P to Q will be contained in the set P $\times$ Q then the relation will be a subset of P $\times$ Q.

Complete step-by-step solution:
Here we have been provided with a relation from P to Q and we are asked to choose the correct option relating to this relation. First we need to understand the term ‘relation’.
In mathematics, if we have two sets P and Q then the binary relation R is defined to be a subset of P $\times$ Q from a set P to Q. Let m and n are the elements in P and Q respectively such that (m, n) $\in$ R and R is a subset of P $\times$ Q, then m is related to n by R, i.e. a R b.
Let us assume the set P = {1, 2, 6} and set Q = {a, b, c} and the relation is defined as,
$\Rightarrow$ R = {(1, a), (1, b), (2, a), (6, c)}
The set P $\times$ Q will be given as,
$\Rightarrow$ P $\times$ Q = {(1, a), (1, b), (1, c), (2, a), (2, b), (2, c), (6, a), (6, b), (6, c)}
Clearly we can see that every element of relation R is present in the set P $\times$ Q, so R is a subset of P $\times$ Q.
Hence, option (d) is the correct answer.

Note: Note that it may be possible that the relation R becomes equal to the set P $\times$ Q, in such a case the relation will become equivalent set of P $\times$ Q. Two equivalent set contains same number of elements and it is not necessary that the elements must be the same. The relation R will not contain any element that may not be present in P $\times$ Q. Option (b) and (c) can also be correct in particular cases but not all the cases and therefore option (d) is the most suitable answer.