p ∧ q ≡ q ∧ pp ∨ q ≡ q ∨ p
To prove the given statements using logical equivalences, we can apply the basic laws of propositional logic
To prove the given statements using logical equivalences, we can apply the basic laws of propositional logic. Here’s the step-by-step breakdown:
Statement 1: p ∧ q ≡ q ∧ p
First, let’s simplify both sides of the equation using the commutative property of conjunction (p ∧ q ≡ q ∧ p):
p ∧ q ≡ q ∧ p
Now, let’s consider the truth table for both sides:
| p | q | p ∧ q | q ∧ p |
|——-|——-|——–|——-|
| true | true | true | true |
| true | false | false | false |
| false | true | false | false |
| false | false | false | false |
As we can see from the truth table, the values of p ∧ q and q ∧ p are always the same for any combination of true/false for p and q. Therefore, p ∧ q ≡ q ∧ p is true.
Statement 2: p ∨ q ≡ q ∨ p
Let’s simplify both sides of the equation using the commutative property of disjunction (p ∨ q ≡ q ∨ p):
p ∨ q ≡ q ∨ p
Now, let’s consider the truth table for both sides:
| p | q | p ∨ q | q ∨ p |
|——-|——-|——–|——-|
| true | true | true | true |
| true | false | true | true |
| false | true | true | true |
| false | false | false | false |
As we can see from the truth table, the values of p ∨ q and q ∨ p are always the same for any combination of true/false for p and q. Therefore, p ∨ q ≡ q ∨ p is true.
In conclusion, both p ∧ q ≡ q ∧ p and p ∨ q ≡ q ∨ p are true based on the logical equivalences used.
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