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Q7:
Why is Axiom 5, in the list of Euclid’s axioms, considered a ‘universal truth’? (Note that the question is not about the fifth postulate.)
Solution :
Step 1: Defining Euclid's Fifth Axiom
Euclid’s fifth axiom in his list of common notions states: "The whole is greater than the part." [As defined in the Elements, Book I, Common Notion 5].
Step 2: Analytical Decomposition
To understand why this is a 'universal truth,' we must evaluate the logical necessity of the statement across mathematical and physical domains:
- Definition of 'Whole': An entity $W$ is defined as the union of all its constituent parts $\{p_1, p_2, p_3, \dots, p_n\}$.
- Definition of 'Part': Any proper subset $P \subset W$ such that $P \neq W$ and $P \neq \emptyset$.
- Logical Axiomatization: If $W = P \cup R$ (where $R$ is the remainder of the whole), and $R$ has a non-zero magnitude (or measure), then the measure of $W$ must exceed the measure of $P$.
Step 3: Verification via Geometric Representation
Consider a line segment $AB$ of length $L$. Let point $C$ lie between $A$ and $B$. By Euclid’s Segment Addition Postulate, $AC + CB = AB$.
In the figure above, the whole segment $AB$ is clearly greater than either of its components $AC$ or $CB$ because $AB = AC + CB$ [Per the definition of betweenness of points].
Step 4: Theoretical Justification of Universality
The axiom is considered a 'universal truth' for the following reasons:
| Criterion | Explanation |
|---|---|
| Self-Evidence | The truth of the statement is immediately apparent to the human intellect without requiring further proof or derivation. |
| Consistency | The axiom holds across all mathematical systems (Arithmetic, Set Theory, Measure Theory) without contradiction. |
| Scope | It is applicable to all physical quantities, including length, area, volume, weight, and abstract sets, making it a foundational common notion. |
Step 5: Conclusion
Euclid's fifth axiom serves as a foundational building block for all measurement. If the whole were not greater than the part, the definitions of addition and subtraction in geometry would collapse, rendering the construction of proofs impossible. Since it is the underlying logic upon which all spatial measurements are based, it remains a self-evident, universal truth.
Final Solution: Euclid’s Fifth Axiom ("The whole is greater than the part") is considered a universal truth because it is a self-evident principle applicable to all magnitudes and sets, serving as a fundamental logical necessity for the consistent definition of addition and comparison within mathematics.
More Questions from Class 9 Mathematics Introduction to Euclid’s Geometry EXERCISE 5.1
- Q1(i): Which of the following statements are true and which are false? Give reasons for your answers. (i) Only one line can pass through a single point.
- Q1(ii): Which of the following statements are true and which are false? Give reasons for your answers. (ii) There are an infinite number of lines which pass through two distinct points.
- Q1(iii): Which of the following statements are true and which are false? Give reasons for your answers. (iii) A terminated line can be produced indefinitely on both the sides.
- Q1(iv): Which of the following statements are true and which are false? Give reasons for your answers. (iv) If two circles are equal, then their radii are equal.
- Q1(v): Which of the following statements are true and which are false? Give reasons for your answers. (v) In Fig. 5.9, if $AB = PQ$ and $PQ = XY$, then $AB = X
- Q2(i): Give a definition for each of the following terms. Are there other terms that need to be defined first? What are they, and how might you define them? (i) parallel lines
- Q2(ii): Give a definition for each of the following terms. Are there other terms that need to be defined first? What are they, and how might you define them? (ii) perpendicular lines
- Q2(iii): Give a definition for each of the following terms. Are there other terms that need to be defined first? What are they, and how might you define them? (iii) line segment
- Q2(iv): Give a definition for each of the following terms. Are there other terms that need to be defined first? What are they, and how might you define them? (iv) radius of a circle
- Q2(v): Give a definition for each of the following terms. Are there other terms that need to be defined first? What are they, and how might you define them? (v) square
- Q3: Consider two ‘postulates’ given below: (i) Given any two distinct points $A$ and $B$, there exists a third point $C$ which is in between $A$ and $B$. (ii) There exist at least three points that are not on the same line. Do these postulates contain any undefined terms? Are these postulates consistent? Do they follow from Euclid’s postulates? Explain.
- Q4: If a point $C$ lies between two points $A$ and $B$ such that $AC = BC$, then prove that $AC = \frac{1}{2}AB$. Explain by drawing the figure.
- Q5: In Question 4, point $C$ is called a mid-point of line segment $AB$. Prove that every line segment has one and only one mid-point.
- Q6: In Fig. 5.10, if $AC = BD$, then prove that $AB = CD$.
CBSE Solutions for Class 9 Mathematics Introduction to Euclid’s Geometry
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