Let
| A: | The food is good |
| B: | The service is excellent |
We discovered in Exercise #22 that
The food is good, but the service is poor.
How, might you ask, does one arrive at this?
In order to negate the implication, we begin by negating the one case in which the implication is false: if A is true, and B is false, then the negation must be true. Hence our target wff should be true when both A and B' are true. Consider
A look at the truth table also suggests this approach:
| A | B | A -> B | ? |
| T | T | T | F |
| T | F | F | T |
| F | T | T | F |
| F | F | T | F |
Only one true, and the rest false. This is similar to the truth table for conjunction making A and B' true simultaneously....
Hence A and B' is the desired negation:
| A | B | A -> B | A and B' |
| T | T | T | F |
| T | F | F | T |
| F | T | T | F |
| F | F | T | F |