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Q4(i):
State whether the following statements are true or false. Give reasons for your answers.
(i) Every natural number is a whole number.
Solution :
Step 1: Define the Mathematical Sets
To evaluate the truth value of the statement, we must first establish the rigorous mathematical definitions of the sets involved.
- Natural Numbers ($\mathbb{N}$): This is the set of all counting numbers starting from $1$ and extending to infinity.
Mathematically, $\mathbb{N} = \{1, 2, 3, 4, \dots\}$ - Whole Numbers ($\mathbb{W}$): This is the set of all natural numbers along with the number zero ($0$).
Mathematically, $\mathbb{W} = \{0, 1, 2, 3, 4, \dots\}$
Step 2: Analyze the Set Relationship
By comparing the elements of both sets, we can observe the structural relationship between them.
The set of whole numbers can be expressed as the union of the set containing zero and the set of natural numbers:
$\mathbb{W} = \{0\} \cup \mathbb{N}$
[By the definition of subsets in set theory], if every element of set $A$ is also an element of set $B$, then $A$ is a subset of $B$ ($A \subseteq B$). Since every element in $\mathbb{N}$ ($1, 2, 3, \dots$) is explicitly contained within $\mathbb{W}$, we can state that $\mathbb{N}$ is a proper subset of $\mathbb{W}$ ($\mathbb{N} \subset \mathbb{W}$).
Step 3: Visual Representation via Euler-Venn Diagram
The following diagram illustrates the subset relationship, demonstrating that the boundary of Natural Numbers is entirely enclosed within the boundary of Whole Numbers.
Step 4: Formulate the Logical Conclusion
Because the collection of whole numbers contains all the natural numbers in addition to zero, there is no natural number that is not also a whole number. Therefore, the premise that "every natural number is a whole number" is logically sound and mathematically verified.
Final Solution: True. The statement is true because the set of whole numbers consists of all natural numbers and the number zero ($\mathbb{W} = \mathbb{N} \cup \{0\}$). Therefore, every natural number inherently belongs to the set of whole numbers.
More Questions from Class 9 Mathematics Number Systems EXERCISE 1.1
- Q1: Is zero a rational number? Can you write it in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$?
- Q2: Find six rational numbers between 3 and 4.
- Q3: Find five rational numbers between $\frac{3}{5}$ and $\frac{4}{5}$.
- Q4(ii): State whether the following statements are true or false. Give reasons for your answers. (ii) Every integer is a whole number.
- Q4(iii): State whether the following statements are true or false. Give reasons for your answers. (iii) Every rational number is a whole number.
CBSE Solutions for Class 9 Mathematics Number Systems
Chapters in CBSE - Class 9 Mathematics
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