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Proof by Contradiction in Discrete mathematics

The notation of proof is known as the key to all mathematics. When we want to say a statement that a property holds for all cases or all numbers with absolute certainty, then we will say it not just because it will be quite nice or sounds convincing if we are able to do this. In the field of mathematics, the various types of proofs come up again and again. Proof by contradiction is one of them. In this section, we are going to use an example to prove the statement with the help of proof by contradiction.

Sometimes it is impossible, or it can be very difficult to prove that a conjecture is true with the help of direct arguments/ direct method. For example: Suppose we want to prove that the "square root of two is irrational". There can be a lot of infinite numbers of rational numbers that might contain a square root of 2, but we cannot direct test or reject them. In place of this, we will do an assumption that root two is rational leads to a contradiction. There is also a powerful tool known as "Proof by Contradiction", which is used to prove that a conjecture is true by the indirect argument.

There are some steps that need to be taken to proof by contradiction, which is described as follows:

Step 1: In the first step, we will assume the opposite of conclusion, which is described as follows:

  1. To prove the statement "the primes are infinite in number", we will assume that the primes are a finite set of size n.
  2. For the statement "if a triangle is scalene, then no two of its angles are congruent", we will assume that the minimum of 2 angles is congruent.

Step 2: As long as the assumption is opposite of our premises, we just begin the use of assumptions to derive new consequences. We need to establish some things for the above two examples, which are described as follows:

  1. For the first example, we will establish that there exists a prime that is not counted in the starting set of n primes.
  2. For the second example, we will be established that the triangle cannot be scalene.

Step 3: It is the last step. Here, we will conclude that our assumption is false and it is the opposite, which means our original conclusion is true.

Now we will understand how this method makes sense. To understand this, we will note that we are creating direct proof of the contrapositive of our original statement. (That means we are proving if not Y, then not X). This is because the contrapositive statements are always logically equivalent, the original then follows.

Note that by the contradiction, we are forced to reject our assumptions. This is because our other steps, which are based on that assumption, are justified and logical. There is only one mistake that we could have made the assumption itself. With the help of indirect proof, it is established that the opposite conclusion is not consistent with the premise. That is the point where the original conclusion must be true.

Definition of Proof by Contradiction:

If we want to prove any statement or something with the help of contradiction, then we will assume that the statement is not true, and after that, we will show that the consequences of the statement are not possible. The consequences are used to contradict either what we have just assumed, or a lot of time, we already know to be true. Sometimes, both things are true, and this is known as the contradiction.

To understand the contradiction, we will take a simple example, in which we will consider "Potter and his parking ticket". As we know that if Potter is unable to pay his parking ticket, then the council will set him a nasty letter. There is one more thing Potter did not get any nasty letters. There will be two cases in which either he will pay his parking ticket or, he did not. With the help of our original information, we know that if Potter did not pay, then he would have got a nasty letter. Therefore, he must pay for his ticket because he did not get a nasty letter.

If we want to formally prove the above example "Potter had paid his ticket" with the help of contradiction, then for this, we will assume that he did not pay his ticket, and then we will deduce that the council must send him a nasty letter. However, we know that Potter's past is particularly pleasant this week and does not contain any nasty letters at all. This is a contradiction, and therefore we can say that our assumption is wrong. The answer to this example was so obvious, but we had taken a long bit to prove this. Sometimes it just takes a long bit in some cases, but in more complicated examples, the proof by contradiction will be very useful to state exactly what we are assuming and where we can find our contradiction.

Example of Proof by Contradiction:

Example 1: In this example, we have to show that the multiplication of an irrational number and a non-zero number will be an irrational number with the help of Proof of contradiction.

Solution: Here, we will prove that the multiplication of an irrational number and a non-zero number will be an irrational number by using the Proof of contradiction.

Statement Comments
For this, we will assume that x is an irrational number and r is a non-zero number.
Let r = m/n, where m and n both are used to indicate the integers. Where m ≠ 0 and n ≠ 0.
Now we will prove that rx is irrational.
We will consider that rx is rational. Here, we take the negation of a statement, which we want to prove.
rx = p/q. where p and q are used to indicate the integers and q ≠ 0. With the help of following the previous statement and also from the rational number definition.
Now we will rearrange the equation rx = p/q, q ≠ 0, and with the help of fact r = m/n, then we will get x = p/rq = np/mq.

Here mq and np are used to indicate the integers, and mq ≠ 0 are used to indicate the rational number.

With the help of using the rational number definition and using the integers.
Hence, we can say that the result is a contradiction because, from the above discussion, we have proved that x is rational, but we have, x is rational with the help of our hypothesis. This is what we know as Proof by contradiction.
Because of this faulty assumption, the contradiction has arisen, which is rx is rational. Hence, rx is irrational. With the help of logical deduction.

Hence, with the help of proof by contradiction method, we have proved the given statement.

Example 2: Suppose there are two statements, p and q.

Here statement p is "x = a/b", where a, and b are used to indicate the co-prime numbers. Statement q is = "2 divides both a, and b".

In this case, we will consider that the statement p is true. We have to also prove that the statement q is also true. Hence, we have arrived at a contradiction because statement q implies the negation of statement p is true.

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