Questions: Select the truth table for the following expression. m ⇒ ∼ n

Select the truth table for the following expression. m ⇒ ∼ n
Transcript text: Select the truth table for the following expression. \[ m \Rightarrow \sim n \]
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Solution

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Solution Steps

To determine the correct truth table for the expression \( m \Rightarrow \sim n \), we need to evaluate the expression for all possible truth values of \( m \) and \( n \). The implication \( m \Rightarrow \sim n \) is true except when \( m \) is true and \( \sim n \) (not \( n \)) is false. We will generate the truth table by iterating through all combinations of truth values for \( m \) and \( n \).

Step 1: Define the Truth Values

We start by defining the possible truth values for \( m \) and \( n \), which are \( \text{True} \) and \( \text{False} \).

Step 2: Calculate \( \sim n \)

For each combination of \( m \) and \( n \), we calculate \( \sim n \) (the negation of \( n \)).

Step 3: Evaluate \( m \Rightarrow \sim n \)

The implication \( m \Rightarrow \sim n \) is true except when \( m \) is true and \( \sim n \) is false. This can be expressed as \( \neg m \lor \sim n \).

Step 4: Construct the Truth Table

We construct the truth table by evaluating the expression for all combinations of \( m \) and \( n \):

\[ \begin{array}{|c|c|c|c|} \hline m & n & \sim n & m \Rightarrow \sim n \\ \hline \text{True} & \text{True} & \text{False} & \text{False} \\ \text{True} & \text{False} & \text{True} & \text{True} \\ \text{False} & \text{True} & \text{False} & \text{True} \\ \text{False} & \text{False} & \text{True} & \text{True} \\ \hline \end{array} \]

Final Answer

The correct truth table for the expression \( m \Rightarrow \sim n \) is:

\[ \boxed{ \begin{array}{|c|c|c|c|} \hline m & n & \sim n & m \Rightarrow \sim n \\ \hline \text{True} & \text{True} & \text{False} & \text{False} \\ \text{True} & \text{False} & \text{True} & \text{True} \\ \text{False} & \text{True} & \text{False} & \text{True} \\ \text{False} & \text{False} & \text{True} & \text{True} \\ \hline \end{array} } \]

Thus, the answer is the second truth table provided in the question.

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