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

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.

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

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)

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

1. For $p = 1$ and $q = 1$: $$p \rightarrow q = 1 \rightarrow 1 = \text{True} \quad (1)$$

2. For $p = 1$ and $q = 0$: $$p \rightarrow q = 1 \rightarrow 0 = \text{False} \quad (0)$$

3. For $p = 0$ and $q = 1$: $$p \rightarrow q = 0 \rightarrow 1 = \text{True} \quad (1)$$

4. For $p = 0$ and $q = 0$: $$p \rightarrow q = 0 \rightarrow 0 = \text{True} \quad (1)$$

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