Questions: Construct a truth table for the given statement. Identify whether the statement is a tautology. [ (-p rightarrow q) wedge p ] Complete the truth table. p q (sim p rightarrow q) wedge p T T square

Construct a truth table for the given statement. Identify whether the statement is a tautology.
[
(-p rightarrow q) wedge p
]

Complete the truth table.

p q (sim p rightarrow q) wedge p
T T square

Transcript text: mylab.pearson.com Lesson 3.3 Pearson MyLab and Mastering e Spring 2025 - Cook .3 Question 14, 3.3.55 Part 1 of 5 Construct a truth table for the given statement. Identify whether the statement is a tautology. \[ (-p \rightarrow q) \wedge p \] Complete the truth table. \begin{tabular}{cc|c} $p$ & $q$ & $(\sim p \rightarrow q) \wedge p$ \\ \hline$T$ & $T$ & $\square$ \end{tabular} example Get more help -

Solution

Solution Steps

Step 1: Evaluate the negation of p

The statement involves $\sim p$, which is the negation of $p$. When $p$ is True, $\sim p$ is False. When $p$ is False, $\sim p$ is True.

Step 2: Evaluate the implication

We have the implication $\sim p \rightarrow q$. An implication is only false when the hypothesis is true and the conclusion is false.

Step 3: Evaluate the conjunction

Finally, we evaluate the conjunction $(\sim p \rightarrow q) \wedge p$. A conjunction is only true when both statements are true.

Final Answer

Here's the completed truth table:

| p | q | ~p | ~p → q | (~p → q) ∧ p | |---|---|---|---|---| | T | T | F | T | T | | T | F | F | T | T | | F | T | T | T | F | | F | F | T | F | F |

The statement is not a tautology because the last column is not always true.

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