Proving the Logical Equivalence between p ∧ (p ∨ q) and p: Step-by-Step Explanation

p ∧ (p ∨ q) ≡ pp ∨ (p ∧ q) ≡ p

To prove that p ∧ (p ∨ q) ≡ p, we need to show that both expressions are logically equivalent, meaning they have the same truth value for all possible truth assignments to the variables p and q

To prove that p ∧ (p ∨ q) ≡ p, we need to show that both expressions are logically equivalent, meaning they have the same truth value for all possible truth assignments to the variables p and q.

Let’s break down the proof step by step:

First, we can simplify the expression p ∧ (p ∨ q) using the distributive law, which states that a ∧ (b ∨ c) is equivalent to (a ∧ b) ∨ (a ∧ c). Applying this law to our expression, we have:

p ∧ (p ∨ q) ≡ (p ∧ p) ∨ (p ∧ q) <-- Distributive law Next, we simplify the terms (p ∧ p) and (p ∧ q) individually. The term (p ∧ p) represents the logical AND operation between two identical propositions, which is always true. Similarly, (p ∧ q) represents the logical AND operation between two propositions. Now, we can substitute these simplified terms back into our expression: (p ∧ p) ∨ (p ∧ q) ≡ p ∨ (p ∧ q) <-- Simplification Finally, we want to simplify the expression p ∨ (p ∧ q). The logical OR operation between a proposition and the logical AND operation between the same proposition and another proposition is equivalent to the original proposition. Using this property, we can simplify the expression: p ∨ (p ∧ q) ≡ p <-- Simplification Therefore, we have proven that p ∧ (p ∨ q) ≡ p. This demonstrates that the expression p ∧ (p ∨ q) is equivalent to p, regardless of the truth values assigned to p and q.

More Answers: