To construct a truth table for the statement [tex]\( p \leftrightarrow (q \vee p) \)[/tex], let's start by breaking it down step-by-step.
1. Identify all possible truth values for [tex]\( p \)[/tex] and [tex]\( q \)[/tex]:
- Each variable can either be True (T) or False (F).
- This gives us 4 combinations to consider: (T, T), (T, F), (F, T), (F, F).
2. Calculate [tex]\( q \vee p \)[/tex] for each combination:
- The expression [tex]\( q \vee p \)[/tex] (q OR p) is True if at least one of [tex]\( q \)[/tex] or [tex]\( p \)[/tex] is True.
3. Calculate [tex]\( p \leftrightarrow (q \vee p) \)[/tex]:
- The expression [tex]\( p \leftrightarrow (q \vee p) \)[/tex] (p if and only if [tex]\( q \vee p \)[/tex]):
- It is True if both sides of the equivalence have the same truth value (both True or both False).
Let's fill in the truth table step-by-step:
[tex]\[
\begin{array}{|c|c|c|c|}
\hline
p & q & q \vee p & p \leftrightarrow (q \vee p) \\
\hline
T & T & \text{True} & \text{True} \\
T & F & \text{True} & \text{True} \\
F & T & \text{True} & \text{False} \\
F & F & \text{False} & \text{True} \\
\hline
\end{array}
\][/tex]
Explanation:
- First Row: [tex]\( p = T \)[/tex], [tex]\( q = T \)[/tex]
- [tex]\( q \vee p \)[/tex] = True (since both are True)
- [tex]\( p \leftrightarrow (q \vee p) \)[/tex] = True (since both are True)
- Second Row: [tex]\( p = T \)[/tex], [tex]\( q = F \)[/tex]
- [tex]\( q \vee p \)[/tex] = True (since p is True)
- [tex]\( p \leftrightarrow (q \vee p) \)[/tex] = True (since both are True)
- Third Row: [tex]\( p = F \)[/tex], [tex]\( q = T \)[/tex]
- [tex]\( q \vee p \)[/tex] = True (since q is True)
- [tex]\( p \leftrightarrow (q \vee p) \)[/tex] = False (since p is False and [tex]\( q \vee p \)[/tex] is True)
- Fourth Row: [tex]\( p = F \)[/tex], [tex]\( q = F \)[/tex]
- [tex]\( q \vee p \)[/tex] = False (since both are False)
- [tex]\( p \leftrightarrow (q \vee p) \)[/tex] = True (since both are False)
Complete truth table:
[tex]\[
\begin{array}{|c|c|c|c|}
\hline
p & q & q \vee p & p \leftrightarrow (q \vee p) \\
\hline
T & T & \text{True} & \text{True} \\
T & F & \text{True} & \text{True} \\
F & T & \text{True} & \text{False} \\
F & F & \text{False} & \text{True} \\
\hline
\end{array}
\][/tex]
