For each of the following statements, write (T) for true and (F) for false:
iThe smallest integer is zero.
ii-10 is greater than-7.
iii Zero is larger than every negative integer.
iv The sum of two negative integers is a negative integer.
v The sum of a negative integer and a positive integer is always a positive integer.

Answer

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

### Step-by-Step Solution:

1. **Statement i:** The smallest integer is zero.
- **Analysis:** Integers include all negative numbers, zero, and positive numbers. The smallest integer is actually negative infinity, as there is no limit to how small an integer can be. Thus, this statement is **False (F)**.

2. **Statement ii:** -10 is greater than -7.
- **Analysis:** On the number line, -10 is to the left of -7, which means -10 is less than -7. Therefore, this statement is **False (F)**.

3. **Statement iii:** Zero is larger than every negative integer.
- **Analysis:** Zero is indeed greater than all negative integers (e.g., -1, -2, -3, etc.). Thus, this statement is **True (T)**.

4. **Statement iv:** The sum of two negative integers is a negative integer.
- **Analysis:** When you add two negative integers, the result is always negative. For example, -2 + (-5) = -7, which is negative. Thus, this statement is **True (T)**.

5. **Statement v:** The sum of a negative integer and a positive integer is always a positive integer.
- **Analysis:** This statement is not always true. For example, if we take -2 (negative integer) and 1 (positive integer), the sum is -1, which is negative. Therefore, this statement is **False (F)**.

### Final Answers:
- i: F
- ii: F
- iii: T
- iv: T
- v: F