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Construct a truth table for the statement [tex]$m \leftrightarrow (n \rightarrow m)$[/tex].

Complete the truth table.

[tex]\[
\begin{tabular}{|c|c|c|c|c|}
\hline
$m$ & $n$ & $n \rightarrow m$ & $m \leftrightarrow (n \rightarrow m)$ \\
\hline
$T$ & $T$ & $T$ & $T$ \\
\hline
$T$ & $F$ & $T$ & $T$ \\
\hline
$F$ & $T$ & $F$ & $F$ \\
\hline
$F$ & $F$ & $T$ & $F$ \\
\hline
\end{tabular}
\][/tex]



Answer :

GinnyAnswer
Certainly! Let's construct the truth table for the statement [tex]\( m \leftrightarrow (n \rightarrow m) \)[/tex].

We first need to understand the logical operations involved:
1. Implication [tex]\( n \rightarrow m \)[/tex]: This is true except when [tex]\( n \)[/tex] is true and [tex]\( m \)[/tex] is false.
2. Biconditional [tex]\( m \leftrightarrow (n \rightarrow m) \)[/tex]: This is true when both sides have the same truth values.

Here's the step-by-step construction:

1. List all combinations of truth values for [tex]\( m \)[/tex] and [tex]\( n \)[/tex].
2. Fill out the column for [tex]\( n \rightarrow m \)[/tex].
3. Determine the truth value of [tex]\( m \leftrightarrow (n \rightarrow m) \)[/tex].

Now let’s fill in the table:

[tex]\[ \begin{array}{|c|c|c|c|} \hline m & n & n \rightarrow m & m \leftrightarrow (n \rightarrow m) \\ \hline T & T & T & T \\ \hline T & F & T & T \\ \hline F & T & F & T \\ \hline F & F & T & F \\ \hline \end{array} \][/tex]

Explanation of each row:

1. When [tex]\( m \)[/tex] is True and [tex]\( n \)[/tex] is True:
- [tex]\( n \rightarrow m \)[/tex] is True (since [tex]\( m \)[/tex] is True).
- [tex]\( m \leftrightarrow (n \rightarrow m) \)[/tex] is True because both sides are True.

2. When [tex]\( m \)[/tex] is True and [tex]\( n \)[/tex] is False:
- [tex]\( n \rightarrow m \)[/tex] is True (implication with False antecedent is True).
- [tex]\( m \leftrightarrow (n \rightarrow m) \)[/tex] is True because both sides are True.

3. When [tex]\( m \)[/tex] is False and [tex]\( n \)[/tex] is True:
- [tex]\( n \rightarrow m \)[/tex] is False (True antecedent and False consequent).
- [tex]\( m \leftrightarrow (n \rightarrow m) \)[/tex] is True because both sides are False.

4. When [tex]\( m \)[/tex] is False and [tex]\( n \)[/tex] is False:
- [tex]\( n \rightarrow m \)[/tex] is True (implication with False antecedent is True).
- [tex]\( m \leftrightarrow (n \rightarrow m) \)[/tex] is False because one side is False and the other is True.

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