p ∧ ¬p ≡ Fp ∨ ¬p ≡ T
To understand the given logical equivalence, we need to break it down step by step
To understand the given logical equivalence, we need to break it down step by step.
1. p ∧ ¬p:
This is a logical statement that involves both p and its negation, ¬p, being connected by the logical operator ∧ (AND). Here’s a truth table to determine the possible outcomes:
| p | ¬p | p ∧ ¬p |
|——-|—–|——–|
| True | False | False |
| False | True | False |
As you can see, in both cases, p and ¬p cannot be simultaneously true, resulting in the entire statement p ∧ ¬p being evaluated as False.
2. Fp ∨ ¬p:
This is another logical statement that involves the logical operator ∨ (OR) connecting Fp (the statement “False p”) and ¬p. Here’s the truth table:
| p | Fp | ¬p | Fp ∨ ¬p |
|——-|—–|—-|———|
| True | False | False | False |
| False | False | True | True |
In this case, Fp represents a statement that is always False, regardless of the truth value of p. As we can see from the truth table, when p is False, the entire statement Fp ∨ ¬p is evaluated as True.
3. Conclusion:
We have determined the truth values of both p ∧ ¬p and Fp ∨ ¬p. According to the given logical equivalence, we find that the two statements are equivalent, as they have the same truth value in all cases:
p ∧ ¬p ≡ Fp ∨ ¬p ≡ T
This means that for any logical value p, the statement p ∧ ¬p is equivalent to Fp ∨ ¬p, and both statements evaluate to True.
More Answers:
Equivalence Proof: Simplifying p ∧ (p ∨ q) ≡ pProving the Equivalence of p ∧ T and pp ∨ F: Truth Table Analysis
Understanding the Equivalence of p ∧ F ≡ Fp ∨ T ≡ T: A Mathematical Analysis