What must the truth values of 𝑝𝑝 and 𝑞𝑞 be in order to make the sentence 𝑝→¬𝑞𝑝→¬𝑞 false?

To determine the truth values of $p$ and $q$ that make the sentence $p \rightarrow \neg q$ false, we will analyze the implications based on logical truth tables.

1. ### Break Down the Problem

We need to identify when the implication $p \rightarrow \neg q$ is false. Recall that an implication $A \rightarrow B$ is only false when $A$ is true and $B$ is false.

2. ### Relevant Concepts

The implication $p \rightarrow \neg q$ can be expressed in terms of its components:

• $p$ is the antecedent (the "if" part).
• $\neg q$ is the consequent (the "then" part).

The truth table for the implication $p \rightarrow \neg q$ is as follows:

3. ### Analysis and Detail

From the truth table:

• The implication $p \rightarrow \neg q$ is false only in the first case where:
  • $p = T$ (true)
  • $q = T$ (true)

Thus, $\neg q$ would be false.

4. ### Verify and Summarize

The only combination of truth values for $p$ and $q$ that makes $p \rightarrow \neg q$ false is:

• $p$ is true
• $q$ is true

Final Answer

The truth values must be $p = T$ and $q = T$ to make the statement $p \rightarrow \neg q$ false.