Proving logical equivalence: p ∧ p ≡ pp ∨ p ≡ p

p ∧ p ≡ pp ∨ p ≡ p

To prove that p ∧ p ≡ pp ∨ p ≡ p, we can use the laws of propositional logic

To prove that p ∧ p ≡ pp ∨ p ≡ p, we can use the laws of propositional logic.

1. p ∧ p
This is the conjunction of p with itself. Since this is a tautology, the result is always true.

2. pp ∨ p
This is the disjunction of pp and p. If either pp or p is true, then the whole expression is true.
Since the proposition pp is logically equivalent to p (i.e., they have the same truth value), we can rewrite the expression as:
p ∨ p

3. p
This is the tautology where p is always true.

Therefore, p ∧ p ≡ pp ∨ p ≡ p.

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