Question:
The statement pattern \([p\wedge (q\vee r)]\vee [\sim r\wedge \sim q\wedge p]\) is equivalent to _____
The statement pattern \([p\wedge (q\vee r)]\vee [\sim r\wedge \sim q\wedge p]\) is equivalent to _____
Show Hint
Factor out p and use the law of excluded middle.
Updated On: Nov 12, 2026
- \(q\vee r\)
- \(p\vee r\)
- \(r\)
- \(p\)
Show Solution
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The Correct Option is D
Solution and Explanation
Step 1: Rewrite the second term
\(\sim r\wedge\sim q\wedge p = p\wedge\sim(q\vee r)\) by De Morgan's law.
Step 2: Factor
The pattern becomes \([p\wedge(q\vee r)]\vee[p\wedge\sim(q\vee r)]\)
\(=p\wedge[(q\vee r)\vee\sim(q\vee r)]\).
Step 3: Simplify
\((q\vee r)\vee\sim(q\vee r)\) is a tautology, so the result is \(p\).
Final Answer:
The statement pattern is equivalent to p. \[ \boxed{\text{(D) }p} \]
\(\sim r\wedge\sim q\wedge p = p\wedge\sim(q\vee r)\) by De Morgan's law.
Step 2: Factor
The pattern becomes \([p\wedge(q\vee r)]\vee[p\wedge\sim(q\vee r)]\)
\(=p\wedge[(q\vee r)\vee\sim(q\vee r)]\).
Step 3: Simplify
\((q\vee r)\vee\sim(q\vee r)\) is a tautology, so the result is \(p\).
Final Answer:
The statement pattern is equivalent to p. \[ \boxed{\text{(D) }p} \]
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