Question

The logical expression \( [p \wedge (q \vee r)] \vee [\neg r \wedge \neg q \wedge p] \) is equivalent to

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

Hint:

In logical expressions, factor common variables first to simplify quickly.

Answer 1

The Correct Option is D

Step 1: Take common term \( p \).
\[ [p \wedge (q \vee r)] \vee [\neg r \wedge \neg q \wedge p] = p \wedge [(q \vee r) \vee (\neg r \wedge \neg q)] \] Step 2: Simplify the bracketed expression.
\[ (q \vee r) \vee (\neg r \wedge \neg q) = \text{True} \] Step 3: Apply identity law.
\[ p \wedge \text{True} = p \] Step 4: Conclusion.
The given logical expression is equivalent to \( p \).