Question:
A statement logically equivalent to the statement pattern \(\sim [(p\rightarrow \sim q)\wedge (\sim q\vee p)]\) is
A statement logically equivalent to the statement pattern \(\sim [(p\rightarrow \sim q)\wedge (\sim q\vee p)]\) is
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Rewrite the implication using disjunction and simplify using distributive and complement laws.
Updated On: Nov 12, 2026
- \(p\wedge q\)
- \(p\vee q\)
- \(p\)
- \(q\)
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The Correct Option is D
Solution and Explanation
Step 1: Understanding the Concept
Use \(p\rightarrow\sim q\equiv\sim p\vee\sim q\).
Step 2: Simplify inside the bracket
\((\sim p\vee\sim q)\wedge(p\vee\sim q)\equiv(\sim q)\vee(\sim p\wedge p)\) by the distributive law.
Step 3: Complement law
\(\sim p\wedge p\) is a contradiction, so the bracket equals \(\sim q\).
Step 4: Negate
\(\sim(\sim q)\equiv q\).
Final Answer:
The statement pattern is equivalent to q. \[ \boxed{\text{(D) }q} \]
Use \(p\rightarrow\sim q\equiv\sim p\vee\sim q\).
Step 2: Simplify inside the bracket
\((\sim p\vee\sim q)\wedge(p\vee\sim q)\equiv(\sim q)\vee(\sim p\wedge p)\) by the distributive law.
Step 3: Complement law
\(\sim p\wedge p\) is a contradiction, so the bracket equals \(\sim q\).
Step 4: Negate
\(\sim(\sim q)\equiv q\).
Final Answer:
The statement pattern is equivalent to q. \[ \boxed{\text{(D) }q} \]
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