In mathematics, we often come across the term theorem. Theorems and proofs are the basics of mathematics. Theorems are relationships between some mathematical expressions or definitions and their properties, whereas proof is set of statements arguing to establish the theorem.

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Mathematical reasoning is the process of finding the proof for a certain mathematical statement by using logic and deductions. There are various types of proofs such as proof by contradiction and proof by contrapositive. These proofs will take an assumption and then prove the statement based on the assumption.

Proof by contradiction is a process of mathematical reasoning to prove a given statement. In a proof by contradiction, the given steps will be followed.

**1. **Take a proposition P to be proved.

**2.** Make an assumption that P is false and P' is true.

3.Deduce that if P' is true, both Q and Q' for a proposition Q is true.

4.Hence, we get a contradiction.

5. It is proved that the assumption P is false is wrong. P is true.

Let us take an example and understand.

**Statement: **$\sqrt{2}$ is an irrational number.

**Proof:** Assume that $\sqrt{2}$ is a rational number. Then, it can be expressed as a fraction.

$2\ q^2 = p^2$

Since $p^2$ is even p will also be even.

$4\ q^2 = 2(p_{1})^2$

Similarly, it can be proved that b is even numbers.

As both a and b are even numbers then they cannot be relatively prime.**Example 1: **Prove that the negative of an irrational number is irrational.

**Solution:**

Statement: The negative of an irrational number x is irrational.

Proof: Suppose that -x is a rational number.

Then, we will have two integers a and b such that,

$-x$ = $\frac{a}{b}$

Multiplying both sides by -1,

$x$ = $\frac{-a}{b}$

As a is an integer, -a is also an integer. Hence, $\frac{-a}{b}$ is a rational number which is a contradiction. Our assumption is proved

to be wrong.

It is concluded that negative of an irrational number is irrational.**Example 2:** Prove that there is no greatest even integer.

**Solution:** Let us suppose there is a greatest even integer x.

If x is an even integer, then x + 2 is an even integer. Now, (x + 2) > x.

There is an even integer greater than x. Hence, there is a contradiction.

The assumption is wrong and hence, it is proved that there is no greatest even integer.

Proof by contradiction is a process of mathematical reasoning to prove a given statement. In a proof by contradiction, the given steps will be followed.

3.

4.

5.

Let us take an example and understand.

Let us suppose $\sqrt{2}$ = $\frac{p}{q}$ where p and q are relatively prime.

Squaring both sides,

$2$ = $\frac{p^2}{q^2}$

$2\ q^2 = p^2$

Since $p^2$ is even p will also be even.

$4\ q^2 = 2(p_{1})^2$

Similarly, it can be proved that b is even numbers.

As both a and b are even numbers then they cannot be relatively prime.

Statement: The negative of an irrational number x is irrational.

Proof: Suppose that -x is a rational number.

Then, we will have two integers a and b such that,

$-x$ = $\frac{a}{b}$

Multiplying both sides by -1,

$x$ = $\frac{-a}{b}$

As a is an integer, -a is also an integer. Hence, $\frac{-a}{b}$ is a rational number which is a contradiction. Our assumption is proved

to be wrong.

It is concluded that negative of an irrational number is irrational.

If x is an even integer, then x + 2 is an even integer. Now, (x + 2) > x.

There is an even integer greater than x. Hence, there is a contradiction.

The assumption is wrong and hence, it is proved that there is no greatest even integer.

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