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.
