Appendix 1: Proofs in Mathematics | A1.1

Question 1

State whether the following statements are always true, always false or ambiguous. Justify your answers.

(i) All mathematics textbooks are interesting.

(ii) The distance from the Earth to the Sun is approximately $1.5 \times 10^8 \text{ km}$ .

(iii) All human beings grow old.

(iv) The journey from Uttarkashi to Harsil is tiring.

(v) The woman saw an elephant through a pair of binoculars.

Solution

Understand the Question

In mathematical logic, we classify statements based on their truth values:

Always true: A statement that is universally true or an established scientific/mathematical fact under all circumstances.
Always false: A statement that is never true under any circumstances.
Ambiguous: A statement whose truth value cannot be definitively determined because it depends on subjective opinion, personal experience, context, or linguistic interpretation.

(i) All mathematics textbooks are interesting.

Step 1 · Analyze the Statement

Whether a textbook is interesting is a matter of personal opinion and varies from person to person. Since its truth depends on the reader rather than an objective fact, it is not universally true or false.

Answer

(i) Ambiguous

(ii) The distance from the Earth to the Sun is approximately $1.5 \times 10^8 \text{ km}$ .

Step 1 · Analyze the Statement

This is a scientifically verified astronomical fact. The mean distance from the Earth to the Sun is approximately $1.496 \times 10^8 \text{ km} \approx 1.5 \times 10^8 \text{ km}$ .

Answer

(ii) Always true

(iii) All human beings grow old.

Step 1 · Analyze the Statement

Aging is a universal biological process. Every human being who lives naturally undergoes the process of growing old.

Answer

(iii) Always true

(iv) The journey from Uttarkashi to Harsil is tiring.

Step 1 · Analyze the Statement

Fatigue from travel is a subjective experience. It varies from person to person depending on individual physical fitness, mode of travel, and road conditions.

Answer

(iv) Ambiguous

(v) The woman saw an elephant through a pair of binoculars.

Step 1 · Analyze the Statement

This statement describes a specific situation whose truth cannot be verified without context. Furthermore, it is grammatically ambiguous as to whether the woman used binoculars or the elephant possessed binoculars.

Answer

(v) Ambiguous

Common Mistakes

Confusing Subjective Opinions with Facts: Statements involving personal feelings (such as 'interesting' or 'tiring') are subjective and therefore ambiguous, not universally true or false.
Assuming Context for Specific Events: Context-dependent sentences (like an event involving an unknown woman) cannot be assigned a definite universal truth value without specific evidence.

More questions in A1.1

State whether the following statements are always true, always false or ambiguous. Justify your answers.

(i) All mathematics textbooks are interesting.

(ii) The distance from the Earth to the Sun is approximately $1.5 \times 10^8 \text{ km}$ .

(iii) All human beings grow old.

(iv) The journey from Uttarkashi to Harsil is tiring.

(v) The woman saw an elephant through a pair of binoculars.

State whether the following statements are true or false. Justify your answers.

(i) All hexagons are polygons.

(ii) Some polygons are pentagons.

(iii) Not all even numbers are divisible by 2.

(iv) Some real numbers are irrational.

(v) Not all real numbers are rational.

Let $a$ and $b$ be real numbers such that $ab \neq 0$ . Then which of the following statements are true? Justify your answers.

(i) Both $a$ and $b$ must be zero.

(ii) Both $a$ and $b$ must be non-zero.

(iii) Either $a$ or $b$ must be non-zero.

Restate the following statements with appropriate conditions, so that they become true.

(i) If $a^2 > b^2$ , then $a > b$ .

(ii) If $x^2 = y^2$ , then $x = y$ .

(iii) If $(x + y)^2 = x^2 + y^2$ , then $x = 0$ .

(iv) The diagonals of a quadrilateral bisect each other.