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.
More Answers:
Logical Equivalence Proof: Commutative Property of Conjunction and DisjunctionExploring the Laws of Boolean Algebra: Proving Equations using Associative Law
Proving Equivalence using Truth Tables: p ∧ (q ∨ r ) ≡ (p ∧ q) ∨ (p ∧ r ), and p ∨ (q ∧ r ) ≡ (p ∨ q) ∧ (p ∨ r )
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