p ∧ F ≡ Fp ∨ T ≡ T
To prove the given statement using logical equivalences, we’ll break it down into two separate equivalences:
1
To prove the given statement using logical equivalences, we’ll break it down into two separate equivalences:
1. p ∧ F ≡ Fp
2. Fp ∨ T ≡ T
1. p ∧ F ≡ Fp:
To prove this, we’ll use the commutative property of conjunction (p ∧ q ≡ q ∧ p) and the identity property of conjunction (p ∧ T ≡ p).
Starting with the left-hand side (LHS):
p ∧ F
Using the commutative property:
F ∧ p
Using the identity property:
Fp
We have now reached the right-hand side (RHS), Fp.
Therefore, the statement p ∧ F ≡ Fp is true.
2. Fp ∨ T ≡ T:
To prove this, we’ll use the identity property of disjunction (p ∨ T ≡ T).
Starting with the left-hand side (LHS):
Fp ∨ T
Using the commutative property:
T ∨ Fp
Using the identity property:
T
We have reached the right-hand side (RHS), T.
Therefore, the statement Fp ∨ T ≡ T is true.
Based on the proofs of both equivalences, we can conclude that the given statement p ∧ F ≡ Fp ∨ T ≡ T is true.
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