UNDERSTANDING SETS AND SUBSETS OF REAL NUMBERS

By understanding which sets are subsets of types of numbers, we can verify whether statements about the relationships between sets are true or false.

The picture shown below clearly illustrates the subsets of real numbers.

From the above picture, we can list out the following important subsets of real numbers.

(i) Rational numbers

(ii) Irrational numbers

(iii) Whole numbers

(iv) Integers

Example 1 : 

Tell whether the given statement is true or false. Explain your choice.

"All irrational numbers are real numbers"

Answer :

True

Every irrational number is included in the set of real numbers. The irrational numbers are a subset of the real numbers.

Example 2 : 

Tell whether the given statement is true or false. Explain your choice.

"No rational numbers are whole numbers"

Answer :

False

A whole number can be written as a fraction with a denominator of 1, so every whole number is included in the set of rational numbers. The whole numbers are a subset of the rational numbers.

Example 3 : 

Tell whether the given statement is true or false. Explain your choice.

"All rational numbers are integers"

Answer :

False

Every integer is a rational number, but not every rational number is an integer. 

For example, rational numbers such as 3/5 and -5/2 are not integers. 

Example 4 : 

Tell whether the given statement is true or false. Explain your choice.

"Some irrational numbers are integers"

Answer :

False

Real numbers are either rational or irrational numbers.

Integers are rational numbers, so no integers are irrational numbers.

Example 5 : 

Tell whether the given statement is true or false. Explain your choice.

"All whole numbers are rational numbers"

Answer :

True

Whole numbers are a subset of the set of rational numbers and can be written as a ratio of the whole number to 1.

Example 6 : 

Tell whether the given statement is true or false. Explain your choice.

"No irrational numbers are whole numbers"

Answer :

True

All whole numbers are rational numbers.

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