Proving Logical Equivalences: Understanding and Demonstrating Identity, Domination, and Idempotent Laws in Math

p ∧ T ≡ pp ∨ F ≡ p

To understand and prove the given expressions, we will use the following properties of logical equivalences:

1

To understand and prove the given expressions, we will use the following properties of logical equivalences:

1. Identity Law: p ∨ F ≡ p
2. Domination Law: p ∧ T ≡ p
3. Idempotent Law: p ∨ p ≡ p

Now, let’s proceed with proving the given expressions step by step:

Expression 1: p ∧ T ≡ p

Using the Domination Law (Idempotent Law reversed), we have:

p ∧ T ≡ p ∨ F ∧ T
≡ p ∨ (F ∧ T) (Associative Law)
≡ p ∨ F (Identity Law)
≡ p (Identity Law).

Expression 2: pp ∨ F ≡ p

Using the Identity Law, we have:

pp ∨ F ≡ p ∨ p ∨ F
≡ p ∨ F (Idempotent Law)
≡ p (Identity Law).

So we have shown that both the given expressions are true. The logical equivalences hold:

p ∧ T ≡ p

and

pp ∨ F ≡ p.

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