Questions: Complete the following Truth Table p q r p ∧ ~q ∨ ~r TTT (a) TT (b) TFT (c) TF (d) FT (e) FT (f) FFT (g) FF (h)

Transcript text: Complete the following Truth Table \begin{tabular}{|l|c|} \hline $\mathbf{p} \mathbf{q} \mathbf{r} \mathbf{p} \wedge \sim \mathbf{q} \vee \sim \mathbf{r}$ \\ \hline TTT & (a) \\ \hline TT & (b) \\ \hline TFT & (c) \\ \hline TF & (d) \\ \hline FT & (e) \\ \hline FT & (f) \\ \hline FFT & (g) \\ \hline FF & (h) \\ \hline \end{tabular}

Solution

Solution Steps

To complete the truth table, we need to evaluate the logical expression $ \mathbf{p} \wedge \sim \mathbf{q} \vee \sim \mathbf{r} $ for each combination of truth values for $ \mathbf{p} $, $ \mathbf{q} $, and $ \mathbf{r} $. The expression involves logical AND ($\wedge$), logical OR ($\vee$), and logical NOT ($\sim$). We will compute the result for each row by substituting the truth values and applying the logical operations.

Step 1: Define the Logical Expression

The logical expression to evaluate is $ \mathbf{p} \wedge \sim \mathbf{q} \vee \sim \mathbf{r} $. This involves the logical operations AND ($\wedge$), OR ($\vee$), and NOT ($\sim$).

Step 2: Evaluate the Expression for Each Set of Truth Values

We evaluate the expression for each combination of truth values for $ \mathbf{p} $, $ \mathbf{q} $, and $ \mathbf{r} $:

For $ \mathbf{p} = \text{True}, \mathbf{q} = \text{True}, \mathbf{r} = \text{True} $: \[ \text{Result} = (\text{True} \wedge \sim \text{True}) \vee \sim \text{True} = \text{False} \vee \text{False} = \text{False} \]
For $ \mathbf{p} = \text{True}, \mathbf{q} = \text{True}, \mathbf{r} = \text{False} $: \[ \text{Result} = (\text{True} \wedge \sim \text{True}) \vee \sim \text{False} = \text{False} \vee \text{True} = \text{True} \]
For $ \mathbf{p} = \text{True}, \mathbf{q} = \text{False}, \mathbf{r} = \text{True} $: \[ \text{Result} = (\text{True} \wedge \sim \text{False}) \vee \sim \text{True} = \text{True} \vee \text{False} = \text{True} \]
For $ \mathbf{p} = \text{True}, \mathbf{q} = \text{False}, \mathbf{r} = \text{False} $: \[ \text{Result} = (\text{True} \wedge \sim \text{False}) \vee \sim \text{False} = \text{True} \vee \text{True} = \text{True} \]
For $ \mathbf{p} = \text{False}, \mathbf{q} = \text{True}, \mathbf{r} = \text{True} $: \[ \text{Result} = (\text{False} \wedge \sim \text{True}) \vee \sim \text{True} = \text{False} \vee \text{False} = \text{False} \]
For $ \mathbf{p} = \text{False}, \mathbf{q} = \text{True}, \mathbf{r} = \text{False} $: \[ \text{Result} = (\text{False} \wedge \sim \text{True}) \vee \sim \text{False} = \text{False} \vee \text{True} = \text{True} \]
For $ \mathbf{p} = \text{False}, \mathbf{q} = \text{False}, \mathbf{r} = \text{True} $: \[ \text{Result} = (\text{False} \wedge \sim \text{False}) \vee \sim \text{True} = \text{False} \vee \text{False} = \text{False} \]
For $ \mathbf{p} = \text{False}, \mathbf{q} = \text{False}, \mathbf{r} = \text{False} $: \[ \text{Result} = (\text{False} \wedge \sim \text{False}) \vee \sim \text{False} = \text{False} \vee \text{True} = \text{True} \]

Final Answer

\[ \begin{array}{|c|c|} \hline \text{TTT} & \text{False} \\ \hline \text{TTF} & \text{True} \\ \hline \text{TFT} & \text{True} \\ \hline \text{TFF} & \text{True} \\ \hline \text{FTT} & \text{False} \\ \hline \text{FTF} & \text{True} \\ \hline \text{FFT} & \text{False} \\ \hline \text{FFF} & \text{True} \\ \hline \end{array} \]

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