Understanding the Logical Equation p ∧ ¬p ≡ F: A Step by Step Analysis using Basic Laws of Logic

p ∧ ¬p ≡ Fp ∨ ¬p ≡ T

To understand why p ∧ ¬p ≡ F, let’s break it down step by step using the basic laws of logic

To understand why p ∧ ¬p ≡ F, let’s break it down step by step using the basic laws of logic.

First, let’s define the symbols used in the expression:
– ∧ represents the logical AND operator, which returns true if both operands are true, and false otherwise.
– ¬ represents the logical NOT operator, which negates the truth value of its operand.
– ≡ represents the logical equivalence operator, which returns true if both operands have the same truth value, and false otherwise.
– F represents false, which is the logical value for something that is not true.
– T represents true, which is the logical value for something that is true.

Now, let’s work through the expression:

p ∧ ¬p

This means we have two propositions, p and ¬p, and we want to evaluate their conjunction (logical AND).

If p is true, then ¬p is false, since ¬p negates the truth value of p. In this case, p ∧ ¬p becomes true ∧ false, which is false.

If p is false, then ¬p is true. In this case, p ∧ ¬p becomes false ∧ true, which is also false.

Therefore, regardless of the truth value of p, p ∧ ¬p is always false (F).

Now, let’s verify the equivalence with the expression T:

Fp ∨ ¬p

In this case, Fp represents false OR p, which is equivalent to p. Therefore, Fp ∨ ¬p evaluates to p OR ¬p.

When evaluating p OR ¬p, if one of them is true, the entire expression is true. And since p and ¬p are complementary (they have opposite truth values), one of them is always true.

Therefore, p OR ¬p is always true (T).

Hence, p ∧ ¬p ≡ F is equivalent to Fp ∨ ¬p ≡ T.

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