Question

Choose all true statements.

All real numbers are rational numbers.

Some rational numbers are natural numbers.

No real numbers are irrational numbers.

All whole numbers are integers.

Some integers are natural numbers.

No rational numbers are integers.

Answer 1

Set of Numbers

Natural Numbers : The set of Natural Numbers is denoted by $\mathbb{N}$ and is given by

• $\mathbb{N}=\{1,\:2,\:3,\:...\}$

Whole Numbers : The set of Whole Numbers is denoted by $\mathbb{W}$ and is given by

• $\mathbb{W}=\{0,\:1,\:2,\:3,\:...\}$

Integers : The set of Integers is denoted by $\mathbb{Z}$ and is given by

• $\mathbb{Z}=\{0,\:\pm 1,\:\pm 2,\:\pm 3,\:...\}$

Rational Numbers : The set of Rational Numbers is denoted by $\mathbb{Q}$ and is given by

• $\mathbb{Q}=\{x:x=\frac{a}{b},\:a,b\in\mathbb{Z}\:and\:b\neq 0\}$

Irrational Numbers : The set of Irrational Numbers is denoted by $\mathbb{R-Q}$ and is given by

• $\mathbb{R-Q}=\{x:x\in\mathbb{R}\:but\:x\notin\mathbb{Q}\}$

Real Numbers : The set of Real Numbers is denoted by $\mathbb{R}$ and is given by

• $\mathbb{R}=\{x:x\in\mathbb{Q}\:and\:x\in\mathbb{R-Q}\}$

Let us solve the given problem now :

• 1. All real numbers are rational numbers.
• ⇒ This statement is incorrect because there are also those real numbers which are Irrational Numbers.

• 2. Some rational numbers are natural numbers.
• ⇒ This statement is correct because there are also the numbers of the form $\frac{a}{b}$ with $a>0$ and $b=1$, giving only $a$, a non-zero positive integer, i.e. a Natutal Number.

• 3. No real numbers are irrational numbers.
• ⇒ This statement is incorrect because the set of Real Numbers contains both Rational Numbers and Irrational Numbers.

• 4. All whole numbers are integers.
• ⇒ This statement is correct because there the set of Whole Numbers is a subset of Integers.

• 5. Some integers are natural numbers.
• ⇒ This statement is correct because the non-zero positive integers are Natural Numbers.

• 6. No rational numbers are integers.
• ⇒ This statement is incorrect because there are also the numbers of the form $\frac{a}{b}$ with $b=1$, giving only $a$, an integer.

Answer : Correct options

• 2. Some rational numbers are natural numbers.
• 4. All whole numbers are integers.
• 5. Some integers are natural numbers.