Question

complete the truth table. the negation of ( p \to q ) is (
eg (p \to q) ). the statements (
eg (p \to q) ) and ( p land
eg q ) logically equivalent because their truth values are blank for every possible combination of values for ( p ) and ( q ). this means the negation of ( p \to q ) blank ( p land
eg q ). (truth table with columns ( p ), ( q ), (
eg q ), ( p land
eg q ), ( p \to q ), (
eg (p \to q) ) and rows with truth value entries like ( t, t, f, f, t, f ); ( t, f, t, t, f, t ); ( f, t, f, f, t, f ); ( f, f, t, f, t, f ))

Explanation:

Step1: Recall Logical Equivalence

The negation of a conditional \( p \to q \) (which is \(
eg(p \to q) \)) is logically equivalent to \( p \land
eg q \). This is a fundamental rule in propositional logic (from the subfield of Mathematics, specifically Logic/Algebra). To verify, we check the truth values:

• For \( p \to q \), it is false only when \( p \) is true and \( q \) is false (\( p = T, q = F \)). So \(

eg(p \to q) \) is true only when \( p = T, q = F \).

• For \( p \land

eg q \), it is true only when \( p = T \) and \(
eg q = T \) (i.e., \( q = F \)), which is the same condition. Thus, their truth values are identical for all possible values of \( p \) and \( q \), meaning they are logically equivalent.

Answer:

The statements \(
eg(p \to q) \) and \( p \land
eg q \) are logically equivalent because their truth values are identical for all possible combinations of truth values for \( p \) and \( q \).