Question:
The logical statement
\[ [\sim(\sim p \vee q) \vee(p \wedge r) \wedge(\sim q \wedge r)] \]
is equivalent to
The logical statement
\[ [\sim(\sim p \vee q) \vee(p \wedge r) \wedge(\sim q \wedge r)] \]
is equivalent to
Show Hint
Use De Morgan's law to simplify the first part, then distribute and use the absorption law.
Updated On: Nov 13, 2026
- \((\mathrm{p}\wedge \mathrm{r})\wedge \sim \mathrm{q}\)
- \((\sim p\wedge \sim q)\wedge r\)
- \(\sim p\vee r\)
- \((p\wedge \sim q)\vee r\)
Show Solution
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The Correct Option is A
Solution and Explanation
Step 1: Understanding the Concept
Read the statement as \([\sim(\sim p\vee q)\vee(p\wedge r)]\wedge(\sim q\wedge r)\).
By De Morgan, \(\sim(\sim p\vee q)=p\wedge\sim q\).
Step 2: Simplify the First Bracket
\((p\wedge\sim q)\vee(p\wedge r)=p\wedge(\sim q\vee r)\).
Step 3: Combine
The whole statement is \(p\wedge(\sim q\vee r)\wedge\sim q\wedge r\).
Since \(\sim q\wedge r\) already makes \(\sim q\vee r\) true, it reduces to \(p\wedge\sim q\wedge r\).
Step 4: Match
This is \((p\wedge r)\wedge\sim q\), option (A).
Final Answer:
The statement is equivalent to \((p\wedge r)\wedge\sim q\). \[ \boxed{\text{(A) }(p\wedge r)\wedge\sim q} \]
Read the statement as \([\sim(\sim p\vee q)\vee(p\wedge r)]\wedge(\sim q\wedge r)\).
By De Morgan, \(\sim(\sim p\vee q)=p\wedge\sim q\).
Step 2: Simplify the First Bracket
\((p\wedge\sim q)\vee(p\wedge r)=p\wedge(\sim q\vee r)\).
Step 3: Combine
The whole statement is \(p\wedge(\sim q\vee r)\wedge\sim q\wedge r\).
Since \(\sim q\wedge r\) already makes \(\sim q\vee r\) true, it reduces to \(p\wedge\sim q\wedge r\).
Step 4: Match
This is \((p\wedge r)\wedge\sim q\), option (A).
Final Answer:
The statement is equivalent to \((p\wedge r)\wedge\sim q\). \[ \boxed{\text{(A) }(p\wedge r)\wedge\sim q} \]
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