Numbers as Subsets of Real Number Set in a Venn Diagram

» Subsets of the Set of Real Numbers

Rational numbers and irrational numbers taken together, are known as real numbers. Thus, every real number is either a rational or an irrational number. The set of real numbers is denoted by ℝ.

There are many important subsets of set of real numbers which are given below

1. Natural Numbers
The numbers being used in counting as 1, 2, 3, 4, …, called natural numbers.
The set of natural numbers is denoted by ℕ. ℕ={1, 2, 3, 4, 5, 6, …}

2. Whole Numbers
The natural numbers along with number 0 (zero) form the set of whole numbers i.e., 0, 1, 2, 3, , are whole numbers. The set of whole numbers is denoted by 𝕎. 𝕎={0, 1, 2, 3, …}
Set of natural numbers is the proper subset of the set of whole numbers.


3. Integers
The natural numbers, their negatives and zero make the set of integers and it is denoted by ℤ.

ℤ ={…, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, …}

Set ofwhole numbers is the proper subset of integers.

4. Rational Numbers
A number of the form p/q, where p and q both are integers and q≠0, is called a rational number (division by 0 is not permissible).
The set of rational numbers is generally denoted by ℚ.

ℚ={p/q:p,q∈ℤ and q≠0}

All the whole numbers are also rational numbers, since they can be represented as the ratio. E.g.,

The set of integers is the proper subset of the set of rational numbers i.e.,
ℤ⊂ℚ and ℕ⊂ℤ⊂ℚ.

» Numbers as subsets

Notice that any natural number is also an integer.
That is, ℕ⊂ℤ.

Also notice that any integer can be Written as a fraction: 4=4/1, and -15=(-15)/1, for examples. Therefore, any integer is a rational number. That is, ℤ⊂ℚ.

Rational numbers can be used to measure distances, so any rational number is a real number. That is, ℚ⊂ℝ.

Altogether we have that


» Subsets

A set A is said to be a subset of set B if every element of A is also an element of B. In symbols we write AB if aA→a∈B.

We denote set of natural numbers by ℕ
set of integers by ℤ
set of rational numbers by ℚ
set of irrational numbers by 𝕋
set of real numbers by ℝ

We observe that


» Subsets of set of real numbers

Somewhere are many important subsets of ℝ. We give below the names of some of these subsets.

The set of natural numbers ℕ={1, 2, 3, 4, 5, …}
The set of integers ℤ={…, -3, -2, -1, 0, 1, 2, 3, …}
The set of rational numbers ℚ={x∶x=p/q, p,q∈ ℤ and q≠0}

which is read “ℚ is the set of all numbers x such that x equals the quotient p/q, where p and q are integers and q is not zero”. Members of ℚ include -5 (which can be expressed as -5/1), 5/7, 3½ (which can be expressed as 7/2) and -11/3. The set of irrational numbers, denoted by 𝕋, is composed of all other real numbers. Thus 𝕋={x∶x∈ℝ and x∉ℚ}, i.e., all real numbers that are not rational. Members of 𝕋 include √2, √5 and π.

Some of the obvious relations among these subsets are:

ℕ⊂ℤ⊂ℚ, ℚ⊂ℝ, 𝕋⊂ℝ, ℕ⊄𝕋.

» Venn diagrams

Venn Diagrams are the diagrams which represent the relationship between sets. For example, the set of natural numbers ℕ is a subset of set of whole numbers 𝕎 which is a subset of integers ℤ. We can represent this relationship through Venn diagram in the following way.

natural and whole numbers integers in a venn diagram

Example —
The Venn Diagram below represents the relationship between categories of numbers.

Note: The Irrational numbers are represented by the region outside of the Rational numbers, but within the Circle of Real numbers.

numbers as subsets in a venn diagram

Natural #’s are a subset of Whole #’s.
Whole #’s are a subset of Integers.
Integers are a subset of Rational #’s.
Rational #’s are a subset of Real #’s.

Note: If a set is a subset of another set, then the entire circle representing the subset is drawn inside of the circle representing the larger set. This is a nested circle instead of an overlapping circle.

5. Irrational Numbers
A number which cannot be written in the form p/q, where
p and q both are integers and q≠0, is called an irrational number i.e., a number which is not rational is called an irrational number.
The set of irrational numbers is denoted by 𝕋.

𝕋={x∶x∈ℝ and x∉ℚ}

e.g., 0.535335333…, √2, √3 are irrational numbers.

Above subsets can be represented diagramatically as given below

numbers in a venn diagram