Question

The Boolean expression: \[ \neg (p \vee q) \vee (\neg p \wedge q) \] is equivalent to:

• A. \( p \)
• B. \( q \)
• C. \( \neg q \)
• D. \( \neg p \)

Hint:

Using De Morgan's law and distribution simplifies complex Boolean expressions.

Answer 1

The Correct Option is D

Applying De Morgan's Law: \[ \neg (p \vee q) = \neg p \wedge \neg q \] Thus: \[ (\neg p \wedge \neg q) \vee (\neg p \wedge q) \] Using distributive law: \[ \neg p \wedge (\neg q \vee q) \] Since \( \neg q \vee q = 1 \) (tautology): \[ \neg p \wedge 1 = \neg p \] Thus, the Boolean expression simplifies to: \[ \neg p \]

Answer 2

Step 1: Analyzing the given Boolean expression
We are given the Boolean expression: \[ \neg (p \vee q) \vee (\neg p \wedge q) \] Let's break this down and simplify it step by step.

Step 2: Apply De Morgan's Law
First, apply De Morgan's Law to the expression \( \neg (p \vee q) \), which states: \[ \neg (p \vee q) = \neg p \wedge \neg q \] So the expression becomes: \[ (\neg p \wedge \neg q) \vee (\neg p \wedge q) \] This is now a disjunction (OR) of two conjunctions (ANDs).

Step 3: Factor out common terms
Notice that \( \neg p \) is common in both parts of the expression, so we can factor it out: \[ \neg p \wedge (\neg q \vee q) \] Since \( \neg q \vee q \) is a tautology (it is always true), the expression simplifies to: \[ \neg p \wedge \text{True} \] Which simplifies further to: \[ \neg p \]

Step 4: Final Answer
The Boolean expression simplifies to:

\( \neg p \)