Question:

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)) $
$=\sim p \vee(q \wedge r)$
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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.