Question:
The statement \((p\wedge (\sim q))\vee ((\sim p)\wedge q)\vee ((\sim p)\wedge (\sim q))\) is equivalent to :
The statement \((p\wedge (\sim q))\vee ((\sim p)\wedge q)\vee ((\sim p)\wedge (\sim q))\) is equivalent to :
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The first two parts together express exclusive OR. Combine with the third part and compare with the negation of a conjunction.
Updated On: Nov 12, 2026
- \((p\vee q)\)
- \(p\vee (\sim q)\)
- \(\sim p\vee q\)
- \(\sim p\vee \sim q\)
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The Correct Option is D
Solution and Explanation
Step 1: Understanding the Concept
We simplify the compound statement using laws of logic and compare it with the given options.
Step 2: Group the Terms
\((p\wedge\sim q)\vee(\sim p\wedge q)\) is true exactly when \(p\) and \(q\) have different truth values.
Step 3: Add the Third Term
\((\sim p\wedge\sim q)\) is true when both are false.
So the whole statement is true in all cases except when \(p\) and \(q\) are both true.
Step 4: Compare the Options
The statement that is false only when \(p=q=T\) is \(\sim(p\wedge q)\equiv\ \sim p\vee\sim q\).
Final Answer:
The statement is equivalent to not p or not q. \[ \boxed{\text{(D) }\sim p\vee\sim q} \]
We simplify the compound statement using laws of logic and compare it with the given options.
Step 2: Group the Terms
\((p\wedge\sim q)\vee(\sim p\wedge q)\) is true exactly when \(p\) and \(q\) have different truth values.
Step 3: Add the Third Term
\((\sim p\wedge\sim q)\) is true when both are false.
So the whole statement is true in all cases except when \(p\) and \(q\) are both true.
Step 4: Compare the Options
The statement that is false only when \(p=q=T\) is \(\sim(p\wedge q)\equiv\ \sim p\vee\sim q\).
Final Answer:
The statement is equivalent to not p or not q. \[ \boxed{\text{(D) }\sim p\vee\sim q} \]
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