Mastering Various Methods of Mathematical Proofs | Direct Proof and Indirect Proof

Name the two kinds of proofs

In mathematics, there are various kinds of proofs

In mathematics, there are various kinds of proofs. Two commonly used types of proofs are:

1. Direct Proof: This is the most straightforward method of proof, where we start with the given information and use known mathematical principles and logical reasoning to derive a conclusion. It follows a structured format of validating each step by using previously established truths. In a direct proof, we assume the initial statement to be true, and through a series of logical deductions, we arrive at the desired conclusion.

Example of a direct proof:
Statement: For any odd integer n, n^2 – 1 is divisible by 8.
Proof: Let n be an odd integer. Then we can express n as 2k + 1, where k is an integer.
n^2 – 1 = (2k + 1)^2 – 1 = 4k^2 + 4k + 1 – 1 = 4k(k + 1)
Since k(k + 1) is always even, let’s represent it as 2m, where m is an integer.
Therefore, n^2 – 1 = 4(2m) = 8m, which is divisible by 8. Hence, the statement is proved.

2. Indirect Proof (also called Proof by Contradiction): This type of proof assumes the opposite or negation of the statement to be proven and then shows that it leads to a contradiction. By showing that the opposite of the desired conclusion cannot be true, it can be inferred that the original statement must be true.

Example of an indirect proof:
Statement: The square root of 2 is an irrational number.
Proof (by contradiction): Suppose that the square root of 2 is rational. This means it can be expressed as a fraction a/b, where a and b are integers with no common factors other than 1.
Now, let’s assume that a/b is in its simplest form, which implies that a and b have no common factors other than 1.
If we square both sides, we get 2 = a^2/b^2. Rearranging gives a^2 = 2b^2.
This implies that a^2 is an even number, and therefore a must also be even (since the square of an odd number is always odd). Let’s represent a as 2c, where c is an integer.
Substituting back, we get (2c)^2 = 2b^2, which simplifies to 4c^2 = 2b^2 or 2c^2 = b^2.
Now, we see that b^2 is also even, and thus b must be even.
Since both a and b are even, they have a common factor of 2, contradicting our assumption that a/b is in its simplest form.
Therefore, the assumption that the square root of 2 is rational leads to a contradiction. Hence, it is irrational.

Both direct proofs and indirect proofs are valuable methods of mathematical reasoning, and their usage depends on the specific problem or theorem being addressed.

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