State 'T' for true and 'F' for false.
(i) Since, `5 gt 3`. Therefore, `-5 gt -3`.
(ii) The difference between an integer and its additive inverse is always even.
(iii) The sum of three different integers can never be zero.
(iv) All whole numbers are integers.

Options

A. `{:((i),(ii),(iii),"(iv)"),(T,F,T,F):}`
B. `{:((i),(ii),(iii),"(iv)"),(F,F,T,T):}`
C. `{:((i),(ii),(iii),"(iv)"),(F,T,F,T):}`
D. `{:((i),(ii),(iii),"(iv)"),(T,T,F,F):}`

Answer

Let's analyze each statement step by step and determine whether they are true (T) or false (F).

### Statement (i): Since `5 > 3`, therefore `-5 > -3`.

**Step 1:** Understand the comparison of positive and negative integers.
- Positive integers increase in value as you move to the right on a number line, while negative integers decrease in value as you move to the left.

**Step 2:** Compare `5` and `3`.
- Since `5` is greater than `3`, this part of the statement is true.

**Step 3:** Compare `-5` and `-3`.
- On the number line, `-5` is to the left of `-3`. Therefore, `-5` is actually less than `-3`.

**Conclusion:** The statement `-5 > -3` is false.
- **Final Answer:** F

### Statement (ii): The difference between an integer and its additive inverse is always even.

**Step 1:** Define an integer and its additive inverse.
- Let the integer be `x`. Its additive inverse is `-x`.

**Step 2:** Calculate the difference.
- The difference between `x` and `-x` is `x - (-x) = x + x = 2x`.

**Step 3:** Analyze the result.
- The expression `2x` is always even, regardless of whether `x` is even or odd, because it is a multiple of 2.

**Conclusion:** The statement is true.
- **Final Answer:** T

### Statement (iii): The sum of three different integers can never be zero.

**Step 1:** Consider the sum of three different integers.
- Let's take an example: `-3`, `2`, and `1`.

**Step 2:** Calculate the sum.
- The sum is `-3 + 2 + 1 = 0`.

**Step 3:** Analyze the result.
- Since we found a combination of three different integers that sum to zero, the statement is false.

**Conclusion:** The statement is false.
- **Final Answer:** F

### Statement (iv): All whole numbers are integers.

**Step 1:** Define whole numbers and integers.
- Whole numbers are `0, 1, 2, 3, ...` (non-negative).
- Integers include all whole numbers and their negative counterparts: `..., -3, -2, -1, 0, 1, 2, 3, ...`.

**Step 2:** Analyze the relationship.
- Since whole numbers are a subset of integers, all whole numbers are indeed integers.

**Conclusion:** The statement is true.
- **Final Answer:** T

### Summary of Answers:
- (i) F
- (ii) T
- (iii) F
- (iv) T