Questions: p q p → q ------------ T T T F F T F F p q --------- ~ ∧ ∨ → ↔ × ⋱

Transcript text: \begin{tabular}{|c|c|c|c|} \hline$p$ & $q$ & $\square$ & $p \rightarrow q$ \\ \hline T & T & $\square$ & $\square$ \\ \hline T & F & $\square$ & $\square$ \\ \hline F & T & $\square$ & $\square$ \\ \hline F & F & $\square$ & $\square$ \\ \hline \end{tabular} \begin{tabular}{ccc|} \hline$p$ & $q$ & \\ $\sim \square$ & $\square \wedge \square$ & $\square \vee \square$ \\ $\square \rightarrow \square$ & $\square \leftrightarrow \square$ & \\ $\times$ & $\ddots$ \\ $\square$ \end{tabular}

Solution

Solution Steps

To complete the truth table for the conditional statement $ p \rightarrow q $, we need to evaluate the truth value of $ p \rightarrow q $ for all possible combinations of truth values of $ p $ and $ q $. The conditional statement $ p \rightarrow q $ is false only when $ p $ is true and $ q $ is false; otherwise, it is true.

Solution Approach

Create a list of all possible combinations of truth values for $ p $ and $ q $.
For each combination, determine the truth value of $ p \rightarrow q $.
Populate the truth table with the results.

Step 1: Define Truth Values

We define the truth values for $ p $ and $ q $ as follows:

$ p = \text{True} $ (1)
$ p = \text{False} $ (0)
$ q = \text{True} $ (1)
$ q = \text{False} $ (0)

Step 2: Evaluate $ p \rightarrow q $

We evaluate the conditional statement $ p \rightarrow q $ for all combinations of truth values:

For $ p = 1 $ and $ q = 1 $: \[ p \rightarrow q = 1 \rightarrow 1 = \text{True} \quad (1) \]
For $ p = 1 $ and $ q = 0 $: \[ p \rightarrow q = 1 \rightarrow 0 = \text{False} \quad (0) \]
For $ p = 0 $ and $ q = 1 $: \[ p \rightarrow q = 0 \rightarrow 1 = \text{True} \quad (1) \]
For $ p = 0 $ and $ q = 0 $: \[ p \rightarrow q = 0 \rightarrow 0 = \text{True} \quad (1) \]

Step 3: Construct the Truth Table

The truth table for $ p $ and $ q $ along with $ p \rightarrow q $ is as follows:

\[ \begin{array}{|c|c|c|} \hline p & q & p \rightarrow q \\ \hline 1 & 1 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \\ \hline \end{array} \]

Final Answer

The completed truth table shows the results for $ p \rightarrow q $ based on the combinations of truth values for $ p $ and $ q $. The final results are:

For $ (p, q) = (1, 1) $, $ p \rightarrow q = 1 $
For $ (p, q) = (1, 0) $, $ p \rightarrow q = 0 $
For $ (p, q) = (0, 1) $, $ p \rightarrow q = 1 $
For $ (p, q) = (0, 0) $, $ p \rightarrow q = 1 $

Thus, the final answer is: \[ \boxed{ \begin{array}{|c|c|c|} \hline p & q & p \rightarrow q \\ \hline 1 & 1 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \\ \hline \end{array} } \]

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