The chicken came before the egg. No it didn't. prove it!

In my previous article, I discussed the role that statements play in both writing and reading mathematical proofs, as well as how to identify a valid statement and an invalid one. I also mentioned the importance of investigating a conjecture before attempting to write proof. If you missed this article, please click here to review it.

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Direct proof is a method of demonstrating the truth of a proposition in mathematics through the use of logical reasoning and previously established definitions and propositions. Essentially, it involves using logical arguments based on previously proven mathematical principles to prove a statement.

A definition is an agreement that a particular word or phrase will stand for some object, property or other concepts that we expect to refer to often. - Mathematical Reasoning by Ted Sundstrom

This method of proof is commonly used in mathematics, as it allows for a clear and concise demonstration of the validity of a proposition. By relying on already proven concepts and principles, direct proof can provide a solid foundation for the proposition being demonstrated.

To construct a direct proof, a mathematician will typically begin by stating the proposition that they wish to prove, and then proceed to outline the logical steps that will be taken to demonstrate its truth. These steps may involve the use of definitions, previously proven propositions, and other logical reasoning techniques in order to build a convincing argument.

An example would be the statement:

“if x and y are odd integers, then x*y is an odd integer”

Constructing a direct proof for this conditional statement would need a precise definition of what an odd integer is or what it means for an integer to be even.

An integer is said to be even provided that there exists an integer such that:-

a = 2n

While an integer is said to be odd provided that there exists an integer such that:-

a = 2n + 1

Using the above definitions we can prove if 28 is even and if 11 is odd, by doing some simple algebra. thus:

we have: a = 2n, where a = 28 therefore:  28 = 2n, Dividing both sides by 2 we arrive at n = 14

14 is a valid even integer and also exists proving 28 as an even integer.

Proving 11 is an odd integer. we have: a = 2n + 1, where a = 11 therefore: 11 = 2n + 1, (Wonders of algebra happens here) we arrive at n = 5

We have properly defined what it means for an integer to be even or odd. Constructing a direct proof for a conditional statement is a demonstration that the conclusion of the conditional statement follows logically from the hypothesis of the conditional statement.

Theorem: If x and y are odd integers, then x*y is an odd integer

Proof: We assume the x and y are good integers and will prove that x*y is an odd integer. Since x and y are odd. There exists integers m and n such that:

x = 2m + 1, y = 2n + 1

Magical Algebra:  x*y = (2m +1) * (2n +1) x*y = 4mn + 2m + 2n + 1

representing x*y in form of an odd integer(Eg. 2q + 1) x*y = 2(2mn + m + n) + 1

Since m and n are integers and integers are said to be close under addition and multiplication (Another topic to write about). We conclude that (2mn + m + n) is an integer. This means that x*y has been written in form 2q + 1 for some integer q and hence x*y is an odd integer. Consequently, it has been proven that if x and y is an odd integers then x*y is an odd integer.

Conclusion

Proving a statement can be direct in some cases, understanding how to break down a statement and provide meaning to each step can definitely help when trying to know how to approach a direct proof.

“An equation for me has no meaning unless it expresses a thought of God.” Ramanujan