To construct the truth table for the logical statement [tex]\((p \vee q) \leftrightarrow q\)[/tex], let's break it down step by step.
We will examine the possible truth values for [tex]\(p\)[/tex] and [tex]\(q\)[/tex], then calculate [tex]\(p \vee q\)[/tex], and finally determine the truth value of [tex]\((p \vee q) \leftrightarrow q\)[/tex].
1. List all possible truth values for [tex]\(p\)[/tex] and [tex]\(q\)[/tex]:
- [tex]\(p\)[/tex] can be True (T) or False (F).
- [tex]\(q\)[/tex] can be True (T) or False (F).
Hence, there are 4 combinations to consider:
- [tex]\((p=T, q=T)\)[/tex]
- [tex]\((p=T, q=F)\)[/tex]
- [tex]\((p=F, q=T)\)[/tex]
- [tex]\((p=F, q=F)\)[/tex]
2. Determine the truth value of [tex]\(p \vee q\)[/tex] for each combination:
- [tex]\(p \vee q\)[/tex] is True if either [tex]\(p\)[/tex] or [tex]\(q\)[/tex] is True, or both are True.
3. Calculate the truth value of [tex]\((p \vee q) \leftrightarrow q\)[/tex]:
- A biconditional statement [tex]\(A \leftrightarrow B\)[/tex] is True if [tex]\(A\)[/tex] and [tex]\(B\)[/tex] have the same truth value, otherwise it is False.
Let's go through each row:
- For [tex]\(p = T\)[/tex] and [tex]\(q = T\)[/tex]:
- [tex]\(p \vee q = T \vee T = T\)[/tex]
- [tex]\((p \vee q) \leftrightarrow q = T \leftrightarrow T = T\)[/tex]
- For [tex]\(p = T\)[/tex] and [tex]\(q = F\)[/tex]:
- [tex]\(p \vee q = T \vee F = T\)[/tex]
- [tex]\((p \vee q) \leftrightarrow q = T \leftrightarrow F = F\)[/tex]
- For [tex]\(p = F\)[/tex] and [tex]\(q = T\)[/tex]:
- [tex]\(p \vee q = F \vee T = T\)[/tex]
- [tex]\((p \vee q) \leftrightarrow q = T \leftrightarrow T = T\)[/tex]
- For [tex]\(p = F\)[/tex] and [tex]\(q = F\)[/tex]:
- [tex]\(p \vee q = F \vee F = F\)[/tex]
- [tex]\((p \vee q) \leftrightarrow q = F \leftrightarrow F = T\)[/tex]
Now, we can complete the truth table:
[tex]\[
\begin{tabular}{|c|c|c|c|}
\hline
$p$ & $q$ & $p \vee q$ & $(p \vee q) \leftrightarrow q$ \\
\hline
T & T & T & T \\
\hline
T & F & T & F \\
\hline
F & T & T & T \\
\hline
F & F & F & T \\
\hline
\end{tabular}
\][/tex]
This table shows the truth values for [tex]\(p\)[/tex], [tex]\(q\)[/tex], [tex]\(p \vee q\)[/tex], and [tex]\((p \vee q) \leftrightarrow q\)[/tex] for all possible combinations of [tex]\(p\)[/tex] and [tex]\(q\)[/tex].
