Solution to 2.
2. Use a truth table to decide each of the following (and explain how you determined each answer):
a) are the statements consistent?
b) does the first statement imply the second?
c) does the second imply the first?
d) are the statements equivalent?
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| T | T | T | F | F | F | T | F | F | F |
| T | T | F | F | F | T | T | T | F | T |
| T | F | T | F | T | F | F | T | F | T |
| T | F | F | F | T | T | F | T | F | T |
| F | T | T | T | F | F | F | F | F | T |
| F | T | F | T | F | T | F | T | F | T |
| F | F | T | T | T | F | T | T | T | T |
| F | F | F | T | T | T | T | T | T | T |
a) are the statements consistent?
Yes. In the last two rows both statements are T.
b) does the first statement imply the second?
Yes. There is no row in which the first (bold) is T and the second (italic) is false.
c) does the second imply the first?
No. There are rows in which the second (italic) is T but the first (bold) is F. For example, row 2.
d) are the statements equivalent?
No. There are rwos in which they have different truth values. Row 2, for instance.