Equivalence Proof: Simplifying p ∧ (p ∨ q) ≡ p

p ∧ (p ∨ q) ≡ pp ∨ (p ∧ q) ≡ p

To prove that the given statement p ∧ (p ∨ q) ≡ p is true, we need to show that both sides of the equation are equivalent

To prove that the given statement p ∧ (p ∨ q) ≡ p is true, we need to show that both sides of the equation are equivalent.

Starting with the left-hand side (LHS):

p ∧ (p ∨ q)

By the distributive property of logical disjunction (∨) over logical conjunction (∧), we can expand this expression:

= (p ∧ p) ∨ (p ∧ q)

Since p ∧ p is equivalent to p, we can simplify further:

= p ∨ (p ∧ q)

Now, let’s look at the right-hand side (RHS):

pp ∨ (p ∧ q)

Since p repeated (pp) is equivalent to p, we can simplify:

= p ∨ (p ∧ q)

As we can see, the RHS is the same as the simplified expression of the LHS. Therefore, we can conclude that p ∧ (p ∨ q) ≡ p.

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