Question:
The negation of $p{\land}(q \to \sim r)$ is
The negation of $p{\land}(q \to \sim r)$ is
Updated On: May 17, 2024
- $\sim p \lor(p\land r) $
- $\sim p \land (p\land r) $
- $p \lor (p\lor r) $
- $ p \vee (q \wedge r)$
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The Correct Option is B
Solution and Explanation
$ \sim(p \wedge(q \rightarrow \sim r)) =\sim p \vee \sim(q \rightarrow-r) $
$=\sim p \vee(q \wedge \sim(\sim r)) $
$=\sim p \vee(q \wedge r)$
$=\sim p \vee(q \wedge \sim(\sim r)) $
$=\sim p \vee(q \wedge r)$
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Concepts Used:
Statements
A statement is a sentence that is either true or false, but not both true and false simultaneously.
Types of Statements:
Simple Statement
- If a statement cannot be further broken down into various statements, or in simpler words if it is concrete by itself, it is called a Simple Statement.
- Examples include:
- A kite is not a rhombus.
- 15 is an odd number.
Compound Statement
- If a statement can further be broken down into simpler statements so that from a main statement, we can yield more than one statement, then it is called a Compound Statement.
- Consider the statement “10 is non-negative and a multiple of 5” which can be broken down into the statements: “10 is non-negative” and “10 is a multiple of 5”.
If-Then Statements
- If we encounter an if-then statement i.e. ‘if a then b’, then by proving that a is true, b can be proved to be true or if we prove that b is false, then a is also false.
- If we encounter a statement which says ‘a if and only if b’, then we can give reason for such a statement by showing that if a is true, then b is also true and if b is true, then a is also true.
- Example:
- a: 8 is multiple of 64
- b: 8 is a factor of 64
Since one of the given statements i.e. a is true, therefore, a or b is true.