Let `R` be the relation defined on power set of A such that `ARB hArr n(P(A)) = n(P(B))` can be : (where `P(A)` denotes power set of `(A)`

Options

A. Reflexive
B. Symmetric
C. Transitive
D. Equiivalence

Explanation

To solve the problem, we need to analyze the relation \( R \) defined on the power set of a set \( A \). The relation states that for sets \( A \) and \( B \), \( A R B \) if and only if \( n(P(A)) = n(P(B)) \), where \( P(A) \) and \( P(B) \) are the power sets of \( A \) and \( B \), respectively.

### Step-by-Step Solution:

1. **Understanding Power Set**:
The power set \( P(X) \) of a set \( X \) is the set of all subsets of \( X \). If \( X \) has \( n \) elements, then the number of subsets (or the size of the power set) is given by \( n(P(X)) = 2^n \).

2. **Applying to Sets A and B**:
For sets \( A \) and \( B \), let \( n(A) \) be the number of elements in set \( A \) and \( n(B) \) be the number of elements in set \( B \). Therefore, we can express the number of elements in their power sets as:
\[ n(P(A)) = 2^{n(A)} \quad \text{and} \quad n(P(B)) = 2^{n(B)} \]

3. **Setting Up the Relation**:
According to the relation \( A R B \), we have:
\[ n(P(A)) = n(P(B)) \implies 2^{n(A)} = 2^{n(B)} \]

4. **Equating Exponents**:
Since the bases are the same (both are 2), we can equate the exponents:
\[ n(A) = n(B) \]

5. **Reflexivity**:
To check if \( R \) is reflexive, we need to see if \( A R A \) holds:
\[ n(P(A)) = n(P(A)) \quad \text{(True)} \]
Therefore, \( R \) is reflexive.

6. **Symmetry**:
To check if \( R \) is symmetric, assume \( A R B \) holds, which means \( n(A) = n(B) \). Then it follows that:
\[ n(B) = n(A) \implies B R A \]
Hence, \( R \) is symmetric.

7. **Transitivity**:
To check if \( R \) is transitive, assume \( A R B \) and \( B R C \):
\[ n(A) = n(B) \quad \text{and} \quad n(B) = n(C) \]
From this, it follows that:
\[ n(A) = n(C) \implies A R C \]
Thus, \( R \) is transitive.

8. **Conclusion**:
Since \( R \) is reflexive, symmetric, and transitive, we conclude that \( R \) is an equivalence relation on the power set of \( A \).