Show that( ) ( ) ( ) ( ) p q r s and p r q s → → → → → →are not logically equivalent
Question
Show that ((p → q) → (r → s)) and (p → (r → (q → s))) are not logically equivalent.
Solution
To show that the two logical expressions (p → q) → (r → s) and p → (r → (q → s)) are not logically equivalent, we can use a truth table. A truth table lists all possible truth values of the variables and the resulting truth values of the expressions. If there is at least one row in the table where the truth values of the two expressions differ, then the expressions are not logically equivalent.
Here is the truth table for the variables p, q, r, and s:
| p | q | r | s | (p → q) → (r → s) | p → (r → (q → s)) |
|---|---|---|---|---|---|
| T | T | T | T | T | T |
| T | T | T | F | F | F |
| T | T | F | T | T | T |
| T | T | F | F | T | T |
| T | F | T | T | T | T |
| T | F | T | F | F | T |
| T | F | F | T | T | T |
| T | F | F | F | T | T |
| F | T | T | T | T | T |
| F | T | T | F | T | T |
| F | T | F | T | T | T |
| F | T | F | F | T | T |
| F | F | T | T | T | T |
| F | F | T | F | T | T |
| F | F | F | T | T | T |
| F | F | F | F | T | T |
As you can see, there is at least one row (the sixth row) where the truth values of the two expressions differ. Therefore, the two expressions (p → q) → (r → s) and p → (r → (q → s)) are not logically equivalent.
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