Give a counter example to show that if integers are not commutative under subtraction using:
a. 23 and -18 `" "` b. 38 and 54

Answer

To show that integers are not commutative under subtraction, we will provide counterexamples using the pairs of integers given in the question.

### Part (a): Using 23 and -18

1. **Define the integers**: Let \( a = 23 \) and \( b = -18 \).

2. **Calculate \( a - b \)**:
\[ a - b = 23 - (-18) = 23 + 18 = 41 \]

3. **Calculate \( b - a \)**:
\[ b - a = -18 - 23 = -41 \]

4. **Compare the results**:
- From step 2, we have \( a - b = 41 \).
- From step 3, we have \( b - a = -41 \).
- Since \( 41 \neq -41 \), we conclude that \( a - b \neq b - a \).

### Part (b): Using 38 and 54

1. **Define the integers**: Let \( a = 38 \) and \( b = 54 \).

2. **Calculate \( a - b \)**:
\[ a - b = 38 - 54 = -16 \]

3. **Calculate \( b - a \)**:
\[ b - a = 54 - 38 = 16 \]

4. **Compare the results**:
- From step 2, we have \( a - b = -16 \).
- From step 3, we have \( b - a = 16 \).
- Since \( -16 \neq 16 \), we conclude that \( a - b \neq b - a \).

### Conclusion
In both parts (a) and (b), we have shown that subtraction of integers is not commutative because \( a - b \) does not equal \( b - a \).

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