Question

What is the negation of \[p \to \left( {q \wedge r} \right)\]?
A. \[ \sim p \to \sim \left( {q \vee r} \right)\]
B. \[ \sim p \to \sim \left( {q \wedge r} \right)\]
C. \[\left( {q \wedge r} \right) \to p\]
D. \[p \wedge \left( { \sim q \vee \sim r} \right)\]

Answer 1

- Hint: Negation means the contradiction or denial of something. So, here you have to find the opposite statement of the given basically. There are some rules which are applied to find the opposite statement. Use the rules and find the negation.


Formula used
 \[ \sim \left( {p \to q} \right) = \]\[p \wedge \sim q\]
De Morgan’s law: \[ \sim \left( { \sim p \vee q} \right) = \sim \left( { \sim p} \right) \wedge \sim q\]

Complete step by step solution The statement \[p \to q\] means “if \[p\] then \[q\]” or “\[p\] implies \[q\]”. It is a conditional proposition. A conditional statement is false if hypothesis is true and the conclusion is false.
By negating \[p \to q\], we get
\[ \sim \left( {p \to q} \right) = \sim \left( { \sim p \vee q} \right)\]
Using De Morgan’s law, we get
\[ \sim \left( { \sim p \vee q} \right) = \sim \left( { \sim p} \right) \wedge \sim q = p \wedge \sim q\]
\[\therefore \sim \left( {p \to q} \right) = p \wedge \sim q\]
Using this formula, we get
Negation of \[p \to \left( {q \wedge r} \right) = p \wedge \left( { \sim \left( {q \wedge r} \right)} \right)\]
Also, negation of \[q \wedge r\] is \[ \sim \left( {q \wedge r} \right) = \sim q \vee \sim r\]
So, negation of \[p \to \left( {q \wedge r} \right) = p \wedge \left( { \sim q \vee \sim r} \right)\]

Hence the correct answer is option D.

Note: You should be aware about all the rules of proposition. Negation of the statement \[\left( {p \to q} \right)\] is \[p \wedge \sim q\] but negation of \[p \to \left( {q \wedge r} \right) = p \wedge \left( { \sim \left( {q \wedge r} \right)} \right)\]. Don’t mess up these two different formula.