Question

Which of the following is not logically equivalent to the proposition: “A real number is either rational or irrational”?

• A. If a number is neither rational nor irrational then it is not real
• B. If a number is not a rational or not an irrational, then it is not real
• C. If a number is not real, then it is neither rational nor irrational
• D. If a number is real, then it is rational or irrational

Answer 1

The correct option is B

If a number is not a rational or not an irrational, then it is not real

Explanation for the correct options:

Rational number: A rational number is a number that can be expressed as the quotient or fraction $\frac{p}{q}$ of two integers, a numerator $p$ and denominator $q\ne 0$.

Irrational number: All real numbers that aren't rational are referred to as irrational numbers.

Real numbers are a set of rational numbers and irrational numbers.

Logically incorrect statement is “If a number is not a rational or not an irrational, then it is not real”.

Because $\sqrt{2}$ is not a rational number but it is a real number.

Explanation for the incorrect options:

For option (a)

Given “If a number is neither rational nor irrational then it is not real”.

Since real numbers are a set of rational numbers and irrational numbers

Therefore, option (a) is the incorrect option.

For option (c)

Given “If a number is not real, then it is neither rational nor irrational”

Since real numbers are a set of rational numbers and irrational numbers

Therefore, option (c) is the incorrect option.
For option (d)

Given “If a number is real, then it is rational or irrational”

Since real numbers are a set of rational numbers and irrational numbers

Therefore, option (D) is the incorrect option.
Hence, the correct option is option (B).