Appendix 1: Proofs in Mathematics | A1.1

Question 2

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.

Solution

Understand the Question

To determine whether each statement is True or False, we evaluate it using standard mathematical definitions:
- Polygon: A closed plane figure formed by straight line segments.
- Hexagon & Pentagon: 6-sided and 5-sided polygons, respectively.
- Even Number: Any integer divisible by $2$ .
- Real Numbers: The set of numbers comprising all rational and irrational numbers.

(i) All hexagons are polygons.

Step 1 · Verify the Statement

A polygon is a closed two-dimensional figure bounded by straight line segments. A hexagon is a closed six-sided figure with straight line segments, fitting the definition of a polygon.

Therefore, every hexagon is a polygon.

Answer

(i) True

(ii) Some polygons are pentagons.

Step 1 · Verify the Statement

A polygon is a general term for any closed shape with straight sides. A pentagon is a specific five-sided polygon. Since pentagons form a subset of all polygons, some polygons are indeed pentagons.

Answer

(ii) True

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

Step 1 · Verify the Statement

By definition, an even number is an integer of the form $2k$ (where $k$ is an integer) and is divisible by $2$ . Since every even number is divisible by $2$ , the statement is false.

Answer

(iii) False

(iv) Some real numbers are irrational.

Step 1 · Verify the Statement

The set of real numbers is composed of rational numbers and irrational numbers (such as $\sqrt{2}$ and $\pi$ ). Since irrational numbers exist and belong to the set of real numbers, some real numbers are irrational.

Answer

(iv) True

(v) Not all real numbers are rational.

Step 1 · Verify the Statement

Real numbers consist of both rational and irrational numbers. Because irrational numbers (like $\sqrt{3}$ or $\pi$ ) are real numbers that cannot be expressed as rational numbers, not all real numbers are rational.

Answer

(v) True

Common Mistakes

Misinterpreting Quantifiers: Confusing "all" and "some". For example, "Some polygons are pentagons" is true because pentagons are part of the larger family of polygons.
Definition of Even Numbers: Overlooking that divisibility by $2$ is the defining condition of even numbers, making "not all even numbers are divisible by $2$ " strictly false.
Composition of Real Numbers: Forgetting that the real number line includes both rational and irrational numbers, which verifies that not all real numbers are rational.

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.