Construct a truth table for each of the following compound statementswhere p, q, r denote primitive statementa) ד(p ۷ דq) → דp

To construct a truth table for the compound statement $\neg(p \land \neg q) \rightarrow \neg p$, we'll proceed with the following steps:

1. Break Down the Problem

We need to evaluate the logical structure of the statement $\neg(p \land \neg q) \rightarrow \neg p$ using the truth values of $p$, $q$, and $r$.

2. Relevant Concepts

The key logical operators involved are:

• $\neg$ (negation)
• $\land$ (conjunction)
• $\rightarrow$ (implication)

3. Create Columns for Each Statement

We will create a truth table that includes:

• Columns for $p$, $q$, $\neg q$, $p \land \neg q$, $\neg(p \land \neg q)$, $\neg p$, and the final column for $\neg(p \land \neg q) \rightarrow \neg p$.

Truth Table Construction

4. Verify and Summarize

• For the row where $p$ is true and $q$ is true, $\neg(p \land \neg q)$ is true, but $\neg p$ is false, resulting in false for the implication.
• For the row where $p$ is true and $q$ is false, $\neg(p \land \neg q)$ is false, and the implication evaluates to false (since the antecedent is false).
• In the case where $p$ is false and $q$ is true, the implication is true because the antecedent is true and the consequent is true.
• Finally, when both $p$ and $q$ are false, the implication remains true.

Final Answer

The truth table shows that the compound statement $\neg(p \land \neg q) \rightarrow \neg p$ is true in two cases and false in two cases.