Question

Let P be any non-empty set containing p elements. Then, what is the number of relations on P?

• A. $2p$
• B. $2^{p^{2}}$
• C. $p^{2}$
• D. $p^{p}$

Hint:

Total Relations on Set A = $2^{n(A) \times n(A)}$.

Answer 1

The Correct Option is B

Step 1: Concept
A relation on set $P$ is a subset of the Cartesian product $P \times P$.
Step 2: Analysis
If set $P$ has $p$ elements, then the Cartesian product $P \times P$ has $p \times p = p^2$ elements.
Step 3: Calculation
The number of possible relations is equal to the number of subsets of $P \times P$. If a set has $n$ elements, it has $2^n$ subsets. Here, $n = p^2$, so the number of relations is $2^{p^2}$.
Step 4: Conclusion
Hence, the correct answer is $2^{p^2}$.
Final Answer:(B)