1. Simplify the expression [(p ∧ q) ↔ q] by using algebra laws.

2. Simplify the following by using algebra laws ¬(p ∧ q) ∨ (¬p ∧ q).

Let's simplify the expression ( p ∧ q ) ↔ q using logical equivalences.

The biconditional ↔ can be rewritten as a conjunction of two implications:
( p ∧ q ) ↔ q = [( p ∧ q ) → q ] ∧ [ q → ( p ∧ q )] .

The first part ( p ∧ q ) → q is always true by the law of simplification because if both p and q are true, then q is true.

The second part q → ( p ∧ q ) can be written as the contrapositive: ¬ ( p ∧ q ) → ¬ q . But ¬ q is the opposite of the premise q , hence making this false. However, we need to check this further by using simple algebra equivalence which implies it needs both to be true.


Thus, the expression simplifies to the logical equivalence p ↔ T r u e , which simplifies to p .

Now, let's simplify ¬ ( p ∧ q ) ∨ ( ¬ p ∧ q ) :

First, apply De Morgan's Laws to ¬ ( p ∧ q ) :
¬ ( p ∧ q ) = ¬ p ∨ ¬ q .
The expression becomes:
( ¬ p ∨ ¬ q ) ∨ ( ¬ p ∧ q ) .
Now, apply the Distributive Law:

¬ p ∨ ( ¬ q ∨ ( ¬ p ∧ q )) .

Apply the Distributive Law inside:
¬ q ∨ ( ¬ p ∧ q ) is equivalent to ( ¬ q ∨ ¬ p ) ∧ ( ¬ q ∨ q ) .
Since ¬ q ∨ q is a tautology, it simplifies to:
¬ q ∨ ¬ p .
Thus, the whole expression simplifies to:
¬ p ∨ ¬ q .

Answered by JessicaJessy | 2025-07-07