Question

If: $n\left( A \right)=m$ , then number of relations in A are
(a)${{2}^{m}}$
(b)${{2}^{m}}-2$
(c)${{2}^{{{m}^{2}}}}$
(d)None of these

Answer 1

Hint: Try to relate the number of relations of set A to set A by observing the elements of power set $A\times A$. If two sets A and B are containing m and n elements, then $A\times B$ will contain mn elements and a number of elements in the power set will be given by ${{2}^{r}}$. If any set is containing ‘r’ elements in it.

Complete step-by-step answer:

As we know a relation between two sets is a collection of ordered pairs containing one object from each set. It means if the x is from the first set and the object y is from the second set, then the objects are said to be in relation.

So, if set A has m number of elements as given in question $n\left( A \right)=m$ ; then we can get the number of relations given by observing the relations of set A to set A.

As we know $A\times B$ for any set A and B will give all the pairs of the elements A and B and the number of elements of $A\times B$ will be given as ‘mn’, if set A contains m elements and set B contains n elements.

Now, we know the number of relations for any two sets A and B will be given by the total number of ways of selecting the elements of $A\times B$ as per the definition of relation given in starting. And we also know that the total number of ways of selecting all elements of $A\times B$ can be given by power set of it as a power set is a set which represents all the possible number of sets from the given set.

And we know there will be ${{2}^{n}}$ elements in the power set of any set, if set consist of m elements.

Hence, number of relations for $A\times B={{2}^{n\times m}}$

Now, as we need to find relations from set A to set A only.

So, number of elements in $A\times A=m\times m={{m}^{2}}$

Hence, number of elements in power set of $A\times A={{2}^{{{m}^{2}}}}$

So, option (c) is the correct answer.


Note: One may prove the number of relations from A to A by taking m elements in set A and relate it with another m elements of set A and get the permutations of them to get the answer.

One may give an answer as ${{m}^{2}}$, without considering the power set of $\left( A\times A \right)$ , which is wrong. So, take care of it and consider the power set of $A\times A$ , as only the power set will give all the relations from set A to A.