Question

Let \(p, q\) be statements. Which of the following is a negation of \(p \rightarrow q\) ? (1) \(\mathrm{p} \wedge(\sim \mathrm{q})\) \((2) \sim q \rightarrow \sim p\) (3) \(\sim p \vee q\) (4) \(\mathrm{p} \vee(\sim \mathrm{q})\) (5) none of these

Short answer

Option (1) \(p \wedge \sim q\)

Step-by-step solution

Understand the Negation

We need to find the negation of the statement \(p \rightarrow q\). The negation of an implication (if p then q) can be defined as p is true and q is false, which means \(p \wedge \sim q\).

Compare with Options

Compare the derived negation \(p \wedge \sim q\) with the given options: 1) \(p \wedge \sim q\)2) \sim q \rightarrow \sim p\3) \sim p \vee q\4) \(p \vee \sim q\)5) none of these

Identify the Matching Option

Among the given options, option (1) matches the derived negation \(p \wedge \sim q\).

Key concepts

Logical Statements

In logic, a statement is a declarative sentence that is either true or false, but not both. For instance, 'It is raining' can be either true or false depending on the weather.
Each statement is represented by a variable, often by letters such as 'p' or 'q'.
This allows us to formulate logical expressions using these variables.
Logical statements are the building blocks of more complex logical constructs, such as implications and negations.

Implication in Logic

Implication in logic is a fundamental concept. It expresses a conditional statement, often written as \( p \rightarrow q \).
This reads as 'if p, then q'.
Here 'p' is the antecedent (the condition) and 'q' is the consequent (what follows from the condition).
This means that if p is true, then q must also be true. If p is false, the whole implication is considered true, no matter what q is.
These rules underpin conditional statements in logical reasoning. It's crucial in understanding logical arguments, hypothesis testing, and many areas of mathematics.

Negation in Logic

Negation in logic involves reversing the truth value of a statement. If 'p' is a statement, its negation is written as \( \eg p \) or \( \sim p \).
If 'p' is true, \( \sim p \) is false. If 'p' is false, \( \sim p \) is true.
When dealing with implications, the negation transforms the statement. The negation of \( p \rightarrow q \) is \( p \ and \sim q \).
This is understood as 'p is true and q is false'. This transformation is important for constructing logical arguments and understanding constraints and contradictions.