Direct Proof of Mathematical Propositions

Proof by direct argument (or direct proof) is one of the most fundamental proof techniques in mathematics. It involves starting from known facts, definitions, and assumptions and applying logical reasoning to reach a conclusion.

The steps of a direct proof are to:

Assume the given conditions are true.
Use definitions, algebraic manipulations, or logical deductions to step towards the conclusion.
Arrive at the required result and conclude the proof.

For example, suppose we have the following statement

The sum of two odd numbers is even.

To prove this directly, we first apply the definition of an odd number:

An odd number is a number that can be written $n = 2N + 1$ for some $N \in \Z$ . An even number is a number that can be written $n = 2N$ for some $N \in \Z$ .

This means that if we have two numbers $p$ and $q$ then we can write them as

\begin{aligned} p &= 2P + 1, \quad P \in \Z \\ q &= 2Q + 1, \quad Q \in \Z \\ \end{aligned}

And therefore

\begin{aligned} p + q &= (2P + 1) + (2Q + 1), \quad P, Q \in \Z \\ p + q &= 2P + 2Q + 2, \quad P, Q \in \Z \\ p + q &= 2(P + Q + 1), \quad P, Q \in \Z \\ \end{aligned}

Since $P$ and $Q$ are both integers, then so is $R = P + Q + 1$ . This means that $p + q$ can be written as

$p + q = 2R, \quad R \in \Z$

and so it meets the definition of an even number.

We conclude our proof with the initialisation QED, which is short for 'quod erat demonstrandum'. This is a Latin phrase meaning 'that which was to be demonstrated', and signifies that we have completed our proof of the statement we had set out to prove.

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