Show that~ ( ) ( ) p q r and q p r → → → are logically equivalent
Question
Show that~ ( ) ( ) p q r and q p r → → → are logically equivalent
Solution
To show that (p → q) ∨ r and q → (p ∨ r) are logically equivalent, we can use a truth table.
Here are the steps:
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Write down all the possible combinations of truth values for p, q, and r. There are eight possibilities since there are three variables (2^3 = 8).
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Calculate the truth values for (p → q) ∨ r and q → (p ∨ r) for each of these combinations.
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If the truth values for (p → q) ∨ r and q → (p ∨ r) are the same for all combinations, then they are logically equivalent.
Here is the truth table:
| p | q | r | (p → q) ∨ r | q → (p ∨ r) |
|---|---|---|---|---|
| T | T | T | T | T |
| T | T | F | T | T |
| T | F | T | T | T |
| T | F | F | F | F |
| F | T | T | T | T |
| F | T | F | T | T |
| F | F | T | T | T |
| F | F | F | T | T |
As you can see, the truth values for (p → q) ∨ r and q → (p ∨ r) are the same for all combinations of p, q, and r. Therefore, (p → q) ∨ r and q → (p ∨ r) are logically equivalent.
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