Questions: Determine if the two statements are logically equivalent, negations, or neither. p ∨ ~ q ; ~ p ∧ q Negations Neither Logically equivalent

Solution Steps

To determine if the two statements are logically equivalent, negations, or neither, we can use truth tables. We will construct truth tables for both statements and compare their truth values for all possible combinations of truth values for \( p \) and \( q \). If the columns for both statements match exactly, they are logically equivalent. If one is the negation of the other, the columns will be opposite. Otherwise, they are neither.

We are given two logical statements:

1. \( p \vee \sim q \)
2. \( \sim p \wedge q \)

To determine the relationship between these statements, we construct a truth table for all possible truth values of \( p \) and \( q \).

\[ \begin{array}{|c|c|c|c|} \hline p & q & p \vee \sim q & \sim p \wedge q \\ \hline \text{True} & \text{True} & \text{True} & \text{False} \\ \text{True} & \text{False} & \text{True} & \text{False} \\ \text{False} & \text{True} & \text{False} & \text{True} \\ \text{False} & \text{False} & \text{True} & \text{False} \\ \hline \end{array} \]

By examining the truth table, we observe the following:

• The column for \( p \vee \sim q \) is \(\{\text{True, True, False, True}\}\).
• The column for \( \sim p \wedge q \) is \(\{\text{False, False, True, False}\}\).

The truth values of the two statements are opposite for each combination of \( p \) and \( q \). This indicates that the statements are negations of each other.

Final Answer