Which one of the following is/are correct?

(A) Every irrational number is a real number.

(B) Every real number is an irrational number.

(C) The sum or difference of a rational number and an irrational number is an irrational number.

(D) The product or quotient of a non-zero rational number with an irrational number is an irrational number.

Choose the correct answer from the options given below:

Real numbers consist of all numbers on the number line, including both rational numbers (numbers that can be written as fractions) and irrational numbers (numbers that cannot be written as fractions).

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Statement (A): Every irrational number is a real number.

Irrational numbers form a subset of real numbers. Numbers like $\sqrt{2}$, $\pi$, and $\sqrt{5}$ are all irrational and belong to the real number system.

This statement is correct.

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Statement (B): Every real number is an irrational number.

The number $5$ is a real number but not irrational since $5 = \frac{5}{1}$, making it rational.

This statement is incorrect.

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Statement (C): The sum or difference of a rational number and an irrational number is an irrational number.

Consider $2 + \sqrt{3}$ or $\pi - 5$. Both results are irrational.

If the sum were rational, subtracting the original rational number would make an irrational number equal to a rational number, which is impossible.

This statement is correct.

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Statement (D): The product or quotient of a non-zero rational number with an irrational number is an irrational number.

Consider $3 \times \sqrt{2} = 3\sqrt{2}$ or $\frac{\pi}{2}$. Both results are irrational.

The non-zero condition is necessary since $0 \times \sqrt{2} = 0$, which is rational.

This statement is correct.

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Correct statements: (A), (C), and (D)

Incorrect statement: (B)

The answer is: (A), (C) and (D) only