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 ) are logically equivalent because their truth values are blank for ( p ) and ( q ). this means the negation of ( p \to q ) is ( 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 t/f values)

Explanation:

Step1: Recall the truth table for \( p \to q \)

The implication \( p \to q \) is false only when \( p \) is true and \( q \) is false (\( p = T, q = F \)), and true otherwise. So:

• When \( p = T, q = T \): \( p \to q = T \)
• When \( p = T, q = F \): \( p \to q = F \)
• When \( p = F, q = T \): \( p \to q = T \)
• When \( p = F, q = F \): \( p \to q = T \)

Step2: Find the truth values of \(

eg (p \to q) \)
Negation of a statement reverses its truth value. So:

• When \( p \to q = T \), \(

eg (p \to q) = F \)

• When \( p \to q = F \), \(

eg (p \to q) = T \)

Step3: Find the truth values of \(

eg q \)
Negation of \( q \):

• When \( q = T \), \(

eg q = F \)

• When \( q = F \), \(

eg q = T \)

Step4: Find the truth values of \( p \land

eg q \)
Conjunction \( p \land
eg q \) is true only when both \( p \) and \(
eg q \) are true:

• \( p = T, q = T \): \(

eg q = F \), so \( p \land
eg q = F \)

• \( p = T, q = F \): \(

eg q = T \), so \( p \land
eg q = T \)

• \( p = F, q = T \): \(

eg q = F \), so \( p \land
eg q = F \)

• \( p = F, q = F \): \(

eg q = T \), but \( p = F \), so \( p \land
eg q = F \)

Step5: Compare \(

eg (p \to q) \) and \( p \land
eg q \)
From steps 2 and 4, the truth values of \(
eg (p \to q) \) and \( p \land
eg q \) are:

• \( p = T, q = T \): Both \( F \)
• \( p = T, q = F \): Both \( T \)
• \( p = F, q = T \): Both \( F \)
• \( p = F, q = F \): Both \( F \)

So their truth values are identical for all cases 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 the same for all possible truth values of \( p \) and \( q \).