Question:
The logical statement \((p∨q)∧[(\sim p∧q)∨(p∧\sim q)]∧\sim q\) is logically equivalent to ...
The logical statement \((p∨q)∧[(\sim p∧q)∨(p∧\sim q)]∧\sim q\) is logically equivalent to ...
Show Hint
Use the exclusive-or shape of the middle bracket and the final ~q.
Updated On: Oct 1, 2026
- \(p∧\sim q\)
- \(\sim p∧q\)
- \(p∧q\)
- \(\sim p∨\sim q\)
Show Solution
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The Correct Option is A
Solution and Explanation
Step 1: Understanding the Concept:
The middle bracket \((\sim p\wedge q)\vee(p\wedge\sim q)\) is true exactly when \(p\) and \(q\) have different truth values (exclusive OR).
Step 2: Combine With the Last Factor:
If \(\sim q\) is true then \(q\) is false. For the exclusive OR to be true with \(q\) false, \(p\) must be true. So the middle bracket together with \(\sim q\) gives \(p\wedge\sim q\).
Step 3: Algebra Check:
\[ [(\sim p\wedge q)\vee(p\wedge\sim q)]\wedge\sim q=(\sim p\wedge q\wedge\sim q)\vee(p\wedge\sim q\wedge\sim q)=F\vee(p\wedge\sim q)=p\wedge\sim q \]
Step 4: Include the First Factor:
The factor \(p\vee q\) is already true whenever \(p\wedge\sim q\) is true (since \(p\) is true). So it does not change the result.
\[ (p\vee q)\wedge(p\wedge\sim q)=p\wedge\sim q \]
This is option (A).
Final Answer:
The statement is equivalent to \(p\wedge\sim q\), option (A). \[ \boxed{\text{(A) } p\wedge\sim q} \]
The middle bracket \((\sim p\wedge q)\vee(p\wedge\sim q)\) is true exactly when \(p\) and \(q\) have different truth values (exclusive OR).
Step 2: Combine With the Last Factor:
If \(\sim q\) is true then \(q\) is false. For the exclusive OR to be true with \(q\) false, \(p\) must be true. So the middle bracket together with \(\sim q\) gives \(p\wedge\sim q\).
Step 3: Algebra Check:
\[ [(\sim p\wedge q)\vee(p\wedge\sim q)]\wedge\sim q=(\sim p\wedge q\wedge\sim q)\vee(p\wedge\sim q\wedge\sim q)=F\vee(p\wedge\sim q)=p\wedge\sim q \]
Step 4: Include the First Factor:
The factor \(p\vee q\) is already true whenever \(p\wedge\sim q\) is true (since \(p\) is true). So it does not change the result.
\[ (p\vee q)\wedge(p\wedge\sim q)=p\wedge\sim q \]
This is option (A).
Final Answer:
The statement is equivalent to \(p\wedge\sim q\), option (A). \[ \boxed{\text{(A) } p\wedge\sim q} \]
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