Appendix 1: Proofs in Mathematics

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 $1.5 \times 10^8$ 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.

Given that all women are mortal, and suppose that A is a woman, what can we conclude about A?

Given that the product of two rational numbers is rational, and suppose $a$ and $b$ are rationals, what can you conclude about $ab$?

Given that the decimal expansion of irrational numbers is non-terminating, non-recurring, and $\sqrt{17}$ is irrational, what can we conclude about the decimal expansion of $\sqrt{17}$?

Given that $y = x^2 + 6$ and $x = -1$, what can we conclude about the value of $y$?

Given that ABCD is a parallelogram and $\angle B = 80^\circ$. What can you conclude about the other angles of the parallelogram?

Given that PQRS is a cyclic quadrilateral and also its diagonals bisect each other. What can you conclude about the quadrilateral?

Given that $\sqrt{p}$ is irrational for all primes $p$ and also suppose that 3721 is a prime. Can you conclude that $\sqrt{3721}$ is an irrational number? Is your conclusion correct? Why or why not?

Prove that the sum of two consecutive odd numbers is divisible by 4.

Take two consecutive odd numbers. Find the sum of their squares, and then add 6 to the result. Prove that the new number is always divisible by 8.

If $p \ge 5$ is a prime number, show that $p^2 + 2$ is divisible by 3.

[Hint: Use Example 11].

Let $x$ and $y$ be rational numbers. Show that $xy$ is a rational number.

If $a$ and $b$ are positive integers, then you know that $a = bq + r, 0 \le r < b$, where $q$ is a whole number. Prove that $\text{HCF}(a, b) = \text{HCF}(b, r)$.

[Hint : Let $\text{HCF}(b, r) = h$. So, $b = k_1h$ and $r = k_2h$, where $k_1$ and $k_2$ are coprime.]

A line parallel to side BC of a triangle ABC, intersects AB and AC at D and E respectively.

Prove that $\frac{\text{AD}}{\text{DB}} = \frac{\text{AE}}{\text{EC}}$.

State the negations for the following statements :

(i) Man is mortal.

(ii) Line $l$ is parallel to line $m$.

(iii) This chapter has many exercises.

(iv) All integers are rational numbers.

(v) Some prime numbers are odd.

(vi) No student is lazy.

(vii) Some cats are not black.

(viii) There is no real number $x$, such that $\sqrt{x} = -1$.

(ix) 2 divides the positive integer $a$.

(x) Integers $a$ and $b$ are coprime.

In each of the following questions, there are two statements. State if the second is the negation of the first or not.

(i) Mumtaz is hungry. Mumtaz is not hungry.

(ii) Some cats are black. Some cats are brown.

(iii) All elephants are huge. One elephant is not huge.

(iv) All fire engines are red. All fire engines are not red.

(v) No man is a cow. Some men are cows.

EXERCISE A1.5

1. Write the converses of the following statements. (i) If it is hot in Tokyo, then Sharan sweats a lot.

(ii) If Shalini is hungry, then her stomach grumbles.

(iii) If Jaswant has a scholarship, then she can get a degree.

(iv) If a plant has flowers, then it is alive.

(v) If an animal is a cat, then it has a tail.

Write the converses of the following statements. Also, decide in each case whether the converse is true or false.

(i) If triangle ABC is isosceles, then its base angles are equal.

(ii) If an integer is odd, then its square is an odd integer.

(iii) If $x^2 = 1$, then $x = 1$.

(iv) If ABCD is a parallelogram, then AC and BD bisect each other.

(v) If $a$, $b$ and $c$, are whole numbers, then $a + (b + c) = (a + b) + c$.

(vi) If $x$ and $y$ are two odd numbers, then $x + y$ is an even number.

(vii) If vertices of a parallelogram lie on a circle, then it is a rectangle.

Suppose $a + b = c + d$, and $a < c$. Use proof by contradiction to show $b > d$.

Let $r$ be a rational number and $x$ be an irrational number. Use proof by contradiction to show that $r + x$ is an irrational number.

Use proof by contradiction to prove that if for an integer $a$, $a^2$ is even, then so is $a$.

[Hint : Assume $a$ is not even, that is, it is of the form $2n + 1$, for some integer $n$, and then proceed.]

Use proof by contradiction to prove that if for an integer $a$, $a^2$ is divisible by 3, then $a$ is divisible by 3.

Use proof by contradiction to show that there is no value of $n$ for which $6^n$ ends with the digit zero.

Prove by contradiction that two distinct lines in a plane cannot intersect in more than one point.