NIMCET 2009 — Mathematics PYQ

1. Understand the sets involved:

• Domain Set: Set $A$ has $m$ elements. Thus, $n(A)=m$.

• Codomain Set: The target set is $A\times A$.

• The number of elements in $A\times A$ is $n(A)\times n(A)=m\times m=m^2$.

2. Definition of a Relation:

A relation from set $X$ to set $Y$ is defined as any subset of the Cartesian product $X\times Y$.

In this specific problem:

• $X=A$

• $Y=A\times A$

3. Determine the number of elements in the Cartesian product ($X\times Y$):

The total number of ordered pairs in the product $A\times (A\times A)$ is:

4. Calculate the Total Number of Relations:

The total number of relations is the total number of subsets of the Cartesian product $A\times (A\times A)$. If a set has $N$ elements, the number of its subsets is $2^N$.

Conclusion:

Comparing the result $2^{m^3}$ with the given options:

• (a) $m^2$ is incorrect.

• (b) $m^3$ is incorrect (this is the number of elements in the Cartesian product, not the number of relations).

• (c) $m$ is incorrect.

Final Answer:

The correct answer is (d) None of these