To construct a truth table for the expression $p \rightarrow (q \lor \sim p)$, let's break down each component step-by-step and evaluate the expression for all possible truth values of $p$ and $q$.

Step 1: Identify the Components

The expression $p \rightarrow (q \lor \sim p)$ has three parts:

1. $p$
2. $q$
3. $\sim p$ (the negation of $p$)

Step 2: Construct the Inner Expression $q \lor \sim p$

The part inside the parentheses, $q \lor \sim p$, represents a logical "OR" operation between $q$ and the negation of $p$.

Step 3: Evaluate $p \rightarrow (q \lor \sim p)$

The outermost part, $p \rightarrow (q \lor \sim p)$, is a conditional ("if-then") statement.

Truth Table

Let's fill in the truth values step-by-step for all four possible combinations of $p$ and $q$.

Explanation of the Truth Table Rows

1. Row 1: $p = T$, $q = T$

   • $\sim p = F$
   • $q \lor \sim p = T \lor F = T$
   • $p \rightarrow (q \lor \sim p) = T \rightarrow T = T$

2. Row 2: $p = T$, $q = F$

   • $\sim p = F$
   • $q \lor \sim p = F \lor F = F$
   • $p \rightarrow (q \lor \sim p) = T \rightarrow F = F$

3. Row 3: $p = F$, $q = T$

   • $\sim p = T$
   • $q \lor \sim p = T \lor T = T$
   • $p \rightarrow (q \lor \sim p) = F \rightarrow T = T$

4. Row 4: $p = F$, $q = F$

   • $\sim p = T$
   • $q \lor \sim p = F \lor T = T$
   • $p \rightarrow (q \lor \sim p) = F \rightarrow T = T$

Would you like further explanation on each step?

Related Questions:

1. What are the truth values for other logical operators like "AND" and "OR"?
2. How does negation affect the truth value of a statement?
3. Can we simplify $p \rightarrow (q \lor \sim p)$ using logical equivalences?
4. How do truth tables help in understanding logical arguments?
5. What are other common truth tables used in propositional logic?

Tip:

Remember that for a conditional statement $p \rightarrow q$, if $p$ is false, the statement is always true regardless of $q$.