Question

Negate the following sentences. If \(x\) is a rational number and \(x \neq 0,\) then \(\tan (x)\) is not a rational number.

Short answer

The negation of the given statement is: '\(x\) is a rational number and \(x \neq 0,\) and \(\tan (x)\) is a rational number.'

Step-by-step solution

Understanding the Logical Statement

The statement provided is: 'If \(x\) is a rational number and \(x \neq 0,\) then \(\tan (x)\) is not a rational number.' In this statement, P is '\(x\) is a rational number and \(x \neq 0\)' and Q is '\(\tan (x)\) is not a rational number'.

Negating the Implication Statement

The negation of an implication statement 'If P, then Q' is 'P and not Q'. In this case, 'P' is '\(x\) is a rational number and \(x \neq 0\)' and 'not Q' is '\(\tan (x)\) is a rational number'. Combining both 'P' and 'not Q' constructs the negation.

Formulating the Negated Statement

Combining 'P' and 'not Q' as per the rule, the negation of the given statement is: '\(x\) is a rational number and \(x \neq 0,\) and \(\tan (x)\) is a rational number.'