Let total number of relations from set A to set B is 32, then the possible cardinal number of A and B may be.

Options

A. 2,3
B. 5,1
C. 4,1
D. 6,1

Answer

To solve the problem step by step, we need to determine the possible cardinal numbers of sets A and B given that the total number of relations from set A to set B is 32.

### Step-by-Step Solution:

1. **Understanding the Concept of Relations**:
The total number of relations from set A to set B can be calculated using the formula:
\[ \text{Number of relations} = 2^{p \times q} \]
where \( p \) is the number of elements in set A and \( q \) is the number of elements in set B.

2. **Setting Up the Equation**:
We know from the problem that the total number of relations is 32. Thus, we can set up the equation:
\[ 2^{p \times q} = 32 \]

3. **Expressing 32 as a Power of 2**:
We can express 32 as a power of 2:
\[ 32 = 2^5 \]
Therefore, we have:
\[ p \times q = 5 \]

4. **Finding Possible Pairs (p, q)**:
Now, we need to find pairs of integers \( (p, q) \) such that their product equals 5. The possible pairs are:
- \( (1, 5) \)
- \( (5, 1) \)
- \( (5, 1) \) and \( (1, 5) \) are the same in terms of cardinality.

5. **Checking the Given Options**:
Now we will check the provided options to see which pair matches our findings:
- Option 1: \( (2, 3) \) → \( 2 \times 3 = 6 \) (not valid)
- Option 2: \( (5, 1) \) → \( 5 \times 1 = 5 \) (valid)
- Option 3: \( (4, 1) \) → \( 4 \times 1 = 4 \) (not valid)
- Option 4: \( (6, 1) \) → \( 6 \times 1 = 6 \) (not valid)

6. **Conclusion**:
The only valid option that satisfies \( p \times q = 5 \) is:
\[ (5, 1) \]
Thus, the possible cardinal numbers of sets A and B are \( 5 \) and \( 1 \) respectively.

### Final Answer:
The possible cardinal number of A and B is \( (5, 1) \).

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