Which of the following statement/statements is/are true?
I.`(od d)^(od d) = od d`
II. `(od d)^("even") = "even"`

Options

A. a)Only I
B. b)Only II
C. c)Both I and II
D. d)Neither I nor II

Answer

To determine the truth of the statements provided, we need to analyze each statement step by step.

### Statement I: \((od d)^{(od d)} = od d\)

1. **Understanding the terms**:
- "od d" refers to an odd number. Let's denote an odd number as \(x\), where \(x\) can be expressed as \(2n + 1\) for some integer \(n\).

2. **Evaluating the left side**:
- We need to evaluate \(x^x\) (where \(x\) is an odd number).
- The result of raising an odd number to any power (including itself) will always yield an odd number. This is because:
- Odd × Odd = Odd
- Therefore, \(x^x\) will also be odd.

3. **Comparing both sides**:
- The left side \(x^x\) is odd, and the right side is also an odd number (since \(od d\) refers to an odd number).
- Thus, we can conclude that \((od d)^{(od d)} = od d\) is true in the sense that both sides are odd.

### Statement II: \((od d)^{("even")} = "even"\)

1. **Understanding the terms**:
- Here, we are raising an odd number to an even power. Let’s denote the even power as \(2m\) where \(m\) is an integer.

2. **Evaluating the left side**:
- When an odd number is raised to an even power, the result is always even. This is because:
- Odd × Odd = Odd, but when you multiply an odd number by itself an even number of times, the result is still odd.
- Therefore, \((od d)^{("even")}\) will yield an odd number.

3. **Comparing both sides**:
- The left side is odd, while the right side is "even".
- Thus, this statement is false.

### Conclusion:
- Statement I is true.
- Statement II is false.

### Final Answer:
- The true statement is: **I only**.

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