Questions: Construct a truth table for the following statement. [ sim mathrmp wedge sim mathrmq ] Complete the truth table. p q sim p sim q sim p wedge sim q --------------- T T V V nabla T F V V V F T V v v F F V V v

Construct a truth table for the following statement.
[
sim mathrmp wedge sim mathrmq
]

Complete the truth table.

p q sim p sim q sim p wedge sim q
---------------
T T V V nabla
T F V V V
F T V v v
F F V V v

Transcript text: Construct a truth table for the following statement. \[ \sim \mathrm{p} \wedge \sim \mathrm{q} \] Complete the truth table. \begin{tabular}{|c|c|c|c|c|c|} \hline p & q & $\sim p$ & $\sim q$ & \multicolumn{2}{|l|}{$\sim p \wedge \sim q$} \\ \hline T & T & V & V & $\nabla$ & \\ \hline T & F & V & V & V & \\ \hline F & T & V & v & v & \\ \hline F & F & V & V & v & \\ \hline \end{tabular}

Solution

Solution Steps

To construct a truth table for the statement $\sim \mathrm{p} \wedge \sim \mathrm{q}$, we need to evaluate the negation of each variable ($\sim \mathrm{p}$ and $\sim \mathrm{q}$) and then determine the conjunction ($\wedge$) of these negations for all possible truth values of $\mathrm{p}$ and $\mathrm{q}$. The truth table will have columns for $\mathrm{p}$, $\mathrm{q}$, $\sim \mathrm{p}$, $\sim \mathrm{q}$, and $\sim \mathrm{p} \wedge \sim \mathrm{q}$.

Step 1: Define the Variables

Let $ p $ and $ q $ be the two boolean variables. The possible truth values for these variables are:

$ p = \text{True}, q = \text{True} $
$ p = \text{True}, q = \text{False} $
$ p = \text{False}, q = \text{True} $
$ p = \text{False}, q = \text{False} $

Step 2: Calculate Negations

We compute the negations of $ p $ and $ q $:

$ \sim p $ (not $ p $):
- For $ p = \text{True} $, $ \sim p = \text{False} $
- For $ p = \text{False} $, $ \sim p = \text{True} $
$ \sim q $ (not $ q $):
- For $ q = \text{True} $, $ \sim q = \text{False} $
- For $ q = \text{False} $, $ \sim q = \text{True} $

Step 3: Calculate the Conjunction

Next, we compute the conjunction $ \sim p \wedge \sim q $:

For $ p = \text{True}, q = \text{True} $: $ \sim p = \text{False}, \sim q = \text{False} $ → $ \sim p \wedge \sim q = \text{False} $
For $ p = \text{True}, q = \text{False} $: $ \sim p = \text{False}, \sim q = \text{True} $ → $ \sim p \wedge \sim q = \text{False} $
For $ p = \text{False}, q = \text{True} $: $ \sim p = \text{True}, \sim q = \text{False} $ → $ \sim p \wedge \sim q = \text{False} $
For $ p = \text{False}, q = \text{False} $: $ \sim p = \text{True}, \sim q = \text{True} $ → $ \sim p \wedge \sim q = \text{True} $

Step 4: Compile the Truth Table

The complete truth table is as follows:

\[ \begin{array}{|c|c|c|c|c|} \hline p & q & \sim p & \sim q & \sim p \wedge \sim q \\ \hline \text{True} & \text{True} & \text{False} & \text{False} & \text{False} \\ \text{True} & \text{False} & \text{False} & \text{True} & \text{False} \\ \text{False} & \text{True} & \text{True} & \text{False} & \text{False} \\ \text{False} & \text{False} & \text{True} & \text{True} & \text{True} \\ \hline \end{array} \]

Final Answer

The truth table for the statement $ \sim p \wedge \sim q $ is complete, showing that the conjunction is only true when both $ p $ and $ q $ are false. Thus, the final answer is:

\[ \boxed{\text{True only when } p = \text{False} \text{ and } q = \text{False}} \]

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