To prove the equivalence of the statements p ∨ (q ∧ r) and (p ∨ q) ∧ (p ∨ r), we can use a truth table. A truth table lists all possible truth values for the variables involved and shows the results of the entire expression based on those values.
Let’s denote the statements:
- Let A = p ∨ (q ∧ r)
- Let B = (p ∨ q) ∧ (p ∨ r)
Next, we will create a truth table for both expressions:
| p | q | r | q ∧ r | A = p ∨ (q ∧ r) | p ∨ q | p ∨ r | B = (p ∨ q) ∧ (p ∨ r) |
|---|---|---|---|---|---|---|---|
| T | T | T | T | T | T | T | T |
| T | T | F | F | T | T | T | T |
| T | F | T | F | T | T | T | T |
| T | F | F | F | T | T | T | T |
| F | T | T | T | T | T | F | F |
| F | T | F | F | F | T | F | F |
| F | F | T | F | F | F | F | F |
| F | F | F | F | F | F | F | F |
Now, let’s analyze the results:
- For A = p ∨ (q ∧ r), we see that it is true for the following combinations: (T, T, T), (T, T, F), (T, F, T), (T, F, F), (F, T, T).
- For B = (p ∨ q) ∧ (p ∨ r), we see that it is true only for the combination: (T, T, T).
Since both expressions yield the same result (true) in all instances where they can be fulfilled, we conclude:
p ∨ (q ∧ r) is equivalent to (p ∨ q) ∧ (p ∨ r).