Proving Logical Equivalences Using De Morgan’s Laws: A Step-by-Step Guide

¬(p ∧ q) ≡ ¬p ∨ ¬q¬(p ∨ q) ≡ ¬p ∧ ¬q

To prove the two logical equivalences:

1

To prove the two logical equivalences:

1. ¬(p ∧ q) ≡ ¬p ∨ ¬q
2. ¬(p ∨ q) ≡ ¬p ∧ ¬q

We will use the laws of De Morgan’s and the distributive properties of logical connectives.

1. ¬(p ∧ q) ≡ ¬p ∨ ¬q
To prove this equivalence, we will work on each side separately and compare them.

Left-hand side (LHS):
¬(p ∧ q)

Applying De Morgan’s law for conjunction (AND), we have:
¬(p ∧ q) ≡ ¬p ∨ ¬q

Therefore, the left-hand side matches the right-hand side, and the equivalence is proven.

2. ¬(p ∨ q) ≡ ¬p ∧ ¬q
To prove this equivalence, we will again work on each side separately and compare them.

Left-hand side (LHS):
¬(p ∨ q)

Applying De Morgan’s law for disjunction (OR), we have:
¬(p ∨ q) ≡ ¬p ∧ ¬q

Therefore, the left-hand side matches the right-hand side, and the equivalence is proven.

In both cases, the logical equivalences are proven using De Morgan’s laws.

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