Subsets of Set of Real Numbers

Subsets of the Set of Real Numbers

The real number system $\mathbb{R}$ contains many important subsets that you have encountered in earlier classes. These subsets form a hierarchy, and understanding their relationships is essential for working with numbers at the Class 11 level.

The Natural Numbers $\mathbb{N}$

The set of natural numbers is the most basic number system we learn:

These are the counting numbers — positive integers starting from 1. In some contexts, 0 is included, but in the NCERT convention (and most of Class 11 mathematics), $\mathbb{N}$ begins at 1.

The Integers $\mathbb{Z}$

The set of integers extends the natural numbers by including zero and the negatives of all natural numbers:

The letter $\mathbb{Z}$ comes from the German word Zahlen, meaning "numbers." Every natural number is an integer, but not every integer is a natural number — for example, $-5$ is an integer but not a natural number.

The Rational Numbers $\mathbb{Q}$

Rational numbers are numbers that can be expressed as a fraction of two integers, with the denominator non-zero:

Read this as: "$\mathbb{Q}$ is the set of all numbers $x$ such that $x$ equals the quotient $\frac{p}{q}$, where $p$ and $q$ are integers and $q$ is not zero."

The letter $\mathbb{Q}$ stands for "quotient." Some examples of rational numbers:

• $-5$ can be written as $\frac{-5}{1}$ or $\frac{5}{-1}$
• $\frac{5}{7}$ is already in the required form
• $6\frac{1}{2}$ can be expressed as $\frac{13}{2}$
• $\frac{-11}{3}$ is also rational

Every integer is rational (because any integer $n$ can be written as $\frac{n}{1}$), but not every rational number is an integer — $\frac{2}{3}$ is rational but not an integer.

The Irrational Numbers $\mathbb{T}$

Irrational numbers are all real numbers that are not rational. The set of irrational numbers is denoted by $\mathbb{T}$:

In words: $\mathbb{T}$ is the set of all real numbers that are not rational. Some familiar irrational numbers:

• $\sqrt{2}$ (cannot be expressed as $\frac{p}{q}$ — this was proved by the ancient Greeks)
• $\sqrt{5}$
• $\pi$ (the ratio of a circle's circumference to its diameter)

Relationships Among These Subsets

The textbook identifies several obvious relations among these subsets. Each relation is a subset statement, and understanding why each holds (or doesn't hold) is crucial.

Relation 1: $\mathbb{N} \subset \mathbb{Z}$

Every natural number is an integer. If $n \in \mathbb{N}$, then $n$ is one of $1, 2, 3, \ldots$, and all of these appear in $\mathbb{Z}$. However, $\mathbb{Z}$ contains numbers like $-3$ and $0$ that are not in $\mathbb{N}$, so the inclusion is proper.

Relation 2: $\mathbb{Z} \subset \mathbb{Q}$

Every integer $n$ can be written as $\frac{n}{1}$, which satisfies the definition of a rational number. So $\mathbb{Z} \subseteq \mathbb{Q}$. But $\frac{1}{2} \in \mathbb{Q}$ is not an integer, so $\mathbb{Z} \subset \mathbb{Q}$ (proper subset).

Relation 3: $\mathbb{Q} \subset \mathbb{R}$

By definition, rational numbers are a subset of real numbers. Every rational number corresponds to a point on the number line. But there are real numbers (like $\sqrt{2}$ and $\pi$) that are not rational, so $\mathbb{Q} \subset \mathbb{R}$.

Relation 4: $\mathbb{T} \subset \mathbb{R}$

Irrational numbers are defined as real numbers that are not rational, so every irrational number is automatically a real number. Thus $\mathbb{T} \subseteq \mathbb{R}$, and since $\mathbb{Q}$ is non-empty and disjoint from $\mathbb{T}$, the inclusion is proper.

Relation 5: $\mathbb{N} \not\subset \mathbb{T}$

This is a non-inclusion. Natural numbers are rational (each $n$ is $\frac{n}{1}$), so no natural number is irrational. Therefore $\mathbb{N}$ is not a subset of $\mathbb{T}$ — in fact, $\mathbb{N}$ and $\mathbb{T}$ are disjoint sets. …