Question:
The dual of the statement pattern \(\sim p\rightarrow \sim (\sim p\rightarrow q)\) is...
The dual of the statement pattern \(\sim p\rightarrow \sim (\sim p\rightarrow q)\) is...
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First simplify the implication to an OR form, then swap every conjunction with a disjunction.
Updated On: Nov 12, 2026
- \(p\vee (p\rightarrow \sim q)\)
- \(\sim p\vee (p\rightarrow \sim q)\)
- \(p\wedge (p\rightarrow \sim q)\)
- \(p\wedge (\sim p\rightarrow \sim q)\)
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The Correct Option is C
Solution and Explanation
Step 1: Understanding the Concept
The dual of a statement is obtained by replacing \(\wedge\) by \(\vee\) and \(\vee\) by \(\wedge\), keeping the negations as they are.
First the statement is simplified using \(a\rightarrow b\equiv \sim a\vee b\).
Step 2: Simplify the statement
\(\sim p\rightarrow \sim(\sim p\rightarrow q)\equiv p\vee \sim(p\vee q)\).
By De Morgan's law, \(\sim(p\vee q)\equiv \sim p\wedge \sim q\).
So the statement is \(p\vee(\sim p\wedge \sim q)\).
Step 3: Take the dual
Interchange \(\vee\) and \(\wedge\): the dual is \(p\wedge(\sim p\vee \sim q)\).
Step 4: Compare the options
Since \(p\rightarrow \sim q\equiv \sim p\vee \sim q\), the dual equals \(p\wedge(p\rightarrow \sim q)\), which matches the third option.
Final Answer:
The dual of the given statement pattern is p AND (p implies not q). \[ \boxed{\text{(C) }p\wedge (p\rightarrow \sim q)} \]
The dual of a statement is obtained by replacing \(\wedge\) by \(\vee\) and \(\vee\) by \(\wedge\), keeping the negations as they are.
First the statement is simplified using \(a\rightarrow b\equiv \sim a\vee b\).
Step 2: Simplify the statement
\(\sim p\rightarrow \sim(\sim p\rightarrow q)\equiv p\vee \sim(p\vee q)\).
By De Morgan's law, \(\sim(p\vee q)\equiv \sim p\wedge \sim q\).
So the statement is \(p\vee(\sim p\wedge \sim q)\).
Step 3: Take the dual
Interchange \(\vee\) and \(\wedge\): the dual is \(p\wedge(\sim p\vee \sim q)\).
Step 4: Compare the options
Since \(p\rightarrow \sim q\equiv \sim p\vee \sim q\), the dual equals \(p\wedge(p\rightarrow \sim q)\), which matches the third option.
Final Answer:
The dual of the given statement pattern is p AND (p implies not q). \[ \boxed{\text{(C) }p\wedge (p\rightarrow \sim q)} \]
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