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Q4(iii):
State whether the following statements are true or false. Give reasons for your answers.
(iii) Every rational number is a whole number.
Solution :
Step 1: Formal Definitions of the Number Sets
To evaluate the validity of the statement, we must first establish the rigorous mathematical definitions of the sets involved:
- Rational Numbers ($\mathbb{Q}$): A rational number is any number that can be expressed in the form $\frac{p}{q}$, where $p$ and $q$ are integers ($\mathbb{Z}$) and $q \neq 0$. Examples include $\frac{1}{2}$, $-\frac{3}{4}$, $5$ (since $5 = \frac{5}{1}$), and $0$.
- Whole Numbers ($\mathbb{W}$): The set of whole numbers consists of all non-negative integers. It is defined as $\mathbb{W} = \{0, 1, 2, 3, \dots\}$.
Step 2: Logical Analysis of the Proposition
The given statement asserts that "Every rational number is a whole number." In set theory notation, this proposition claims that the set of rational numbers is a subset of the set of whole numbers ($\mathbb{Q} \subseteq \mathbb{W}$).
[Per the axioms of set theory], for $\mathbb{Q} \subseteq \mathbb{W}$ to be true, every single element that exists in $\mathbb{Q}$ must also exist in $\mathbb{W}$. If we can find even one element in $\mathbb{Q}$ that is not in $\mathbb{W}$, the statement is proven false by counterexample.
Step 3: Disproving via Counterexamples
Let us test specific rational numbers against the definition of whole numbers to find a counterexample:
- Fractional Values: Consider the number $\frac{3}{5}$. It is a rational number because it is in the form $\frac{p}{q}$ where $p=3$, $q=5$, and $q \neq 0$. However, $\frac{3}{5} = 0.6$, which is a decimal value and not a non-negative integer. Therefore, $\frac{3}{5} \notin \mathbb{W}$.
- Negative Integers: Consider the number $-4$. It is a rational number because it can be written as $\frac{-4}{1}$. However, whole numbers strictly include only non-negative integers. Therefore, $-4 \notin \mathbb{W}$.
Step 4: Visualizing the Set Hierarchy
The relationship between these number systems is strictly hierarchical in the opposite direction: $\mathbb{W} \subset \mathbb{Q}$. Every whole number is a rational number, but the converse is not true. The Euler diagram below illustrates this structural relationship, proving that $\mathbb{Q}$ encompasses a much larger domain than $\mathbb{W}$.
Step 5: Final Conclusion
Because the set of rational numbers ($\mathbb{Q}$) contains fractions and negative integers—neither of which belong to the set of whole numbers ($\mathbb{W}$)—the proposition fails. The correct logical statement is that every whole number is a rational number, but not vice versa.
Final Solution: False. Rational numbers include fractions (e.g., $\frac{1}{2}$) and negative integers (e.g., $-3$), which are not whole numbers. Therefore, it is incorrect to state that every rational number is a whole number.
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(i): State whether the following statements are true or false. Give reasons for your answers. (i) Every natural number is a whole number.
- Q4(ii): State whether the following statements are true or false. Give reasons for your answers. (ii) Every integer is a whole number.
CBSE Solutions for Class 9 Mathematics Number Systems
Chapters in CBSE - Class 9 Mathematics
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