p ∧ q ≡ q ∧ pp ∨ q ≡ q ∨ p
To prove the equality p ∧ q ≡ q ∧ p, we need to show that both sides of the equation are equivalent, meaning they have the same truth value for all possible truth values of p and q
To prove the equality p ∧ q ≡ q ∧ p, we need to show that both sides of the equation are equivalent, meaning they have the same truth value for all possible truth values of p and q.
To do this, we can construct a truth table and evaluate both sides of the equation for all possible combinations of truth values for p and q:
“`
| p | q | p ∧ q | q ∧ p |
|—|—|——-|——-|
| T | T | T | T |
| T | F | F | F |
| F | T | F | F |
| F | F | F | F |
“`
From the truth table, we can see that for all possible combinations of truth values, p ∧ q is equal to q ∧ p. Hence, we can conclude that p ∧ q ≡ q ∧ p is true.
Now let’s prove the equality p ∨ q ≡ q ∨ p:
Again, we will construct a truth table and evaluate both sides of the equation for all possible combinations of truth values for p and q:
“`
| p | q | p ∨ q | q ∨ p |
|—|—|——-|——-|
| T | T | T | T |
| T | F | T | T |
| F | T | T | T |
| F | F | F | F |
“`
From the truth table, we can see that for all possible combinations of truth values, p ∨ q is equal to q ∨ p. Hence, we can conclude that p ∨ q ≡ q ∨ p is true.
In summary, both equations p ∧ q ≡ q ∧ p and p ∨ q ≡ q ∨ p are true, as demonstrated by the truth tables above.
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