Q1(iii):

State whether the following statements are true or false. Justify your answers. (iii) Every real number is an irrational number.

Step 1: Theoretical Foundation of the Real Number System

To evaluate the statement, we must first establish the rigorous mathematical definition of real numbers. The set of real numbers, denoted by $\mathbb{R}$, is defined as the union of two mutually exclusive and exhaustive subsets: the set of rational numbers ($\mathbb{Q}$) and the set of irrational numbers. [Per the fundamental axioms of set theory and the real number system].

Mathematically, this relationship is expressed as:

$\mathbb{R} = \mathbb{Q} \cup (\mathbb{R} \setminus \mathbb{Q})$

Where:

• $\mathbb{Q}$ represents rational numbers (numbers that can be expressed as $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$).
• $\mathbb{R} \setminus \mathbb{Q}$ represents irrational numbers (numbers that cannot be expressed as a simple fraction, possessing non-terminating and non-repeating decimal expansions).

Step 2: Logical Analysis of the Statement

The given statement asserts: "Every real number is an irrational number."

In logical terms, this translates to the claim that the set of real numbers is entirely equivalent to the set of irrational numbers ($\mathbb{R} = \mathbb{R} \setminus \mathbb{Q}$). For this to be true, the set of rational numbers ($\mathbb{Q}$) would have to be an empty set ($\emptyset$). However, we know that the set of rational numbers is infinitely large. Therefore, the statement contains a fundamental logical fallacy.

Step 3: Proof by Counterexample

To formally disprove a universal affirmative statement ("every $X$ is $Y$"), we only need to provide a single valid counterexample [By the rules of deductive logic and proof by contradiction].

• Let us select the number $4$.
• $4$ is a real number because it represents a continuous quantity that can be plotted on the one-dimensional real number line.
• However, $4$ can be expressed in the fractional form $\frac{4}{1}$, where both $4$ and $1$ are integers, and the denominator is not zero.
• Therefore, $4$ is a rational number, which explicitly means it is not an irrational number.

Step 4: Visualizing the Subsets of Real Numbers

The following Venn diagram illustrates the composition of the real number system, clearly demonstrating that irrational numbers only make up one portion of the entire set.

Final Solution: False. The set of real numbers is comprised of both rational and irrational numbers. Therefore, a real number can be rational, meaning it is incorrect to state that every real number is irrational. For example, $5$ is a real number, but it is a rational number, not an irrational one.