Question

Using truth table verify that: \( (p \wedge q) \vee \sim q \equiv p \vee \sim q \)

Hint:

Use truth tables to verify logical equivalence by comparing output columns for all input combinations.

Answer 1

Construct a truth table for both expressions:
\[\begin{array}{|c|c|c|c|c|c|} \hline \( p \) & \( q \) & \( \sim q \) & \( p \wedge q \) & \( (p \wedge q) \vee \sim q \) & \( p \vee \sim q \) \\ \hline \text{T} & \text{T} & \text{F} & \text{T} & \text{T} & \text{T} \\ \hline \text{T} & \text{F} & \text{T} & \text{F} & \text{T} & \text{T} \\ \hline \text{F} & \text{T} & \text{F} & \text{F} & \text{F} & \text{F} \\ \hline \text{F} & \text{F} & \text{T} & \text{F} & \text{T} & \text{T} \\ \hline \end{array}\]

Step 1: Compute \( \sim q \): Negation of \( q \).

Step 2: Compute \( p \wedge q \): True only when both \( p \) and \( q \) are true.

Step 3: Compute \( (p \wedge q) \vee \sim q \): OR of \( p \wedge q \) and \( \sim q \).

Step 4: Compute \( p \vee \sim q \): OR of \( p \) and \( \sim q \).
The columns for \( (p \wedge q) \vee \sim q \) and \( p \vee \sim q \) are identical, proving equivalence.
Answer: \( (p \wedge q) \vee \sim q \equiv p \vee \sim q \).