How To Use A Direct Proof To Prove A Math Statement: Example With Odd Integer Multiplication

Direct Proof

a sequence of statements which are either givens or deductions from previous statements, and whose last statement is the conclusion to be proved

A direct proof is a method of proving a statement by logically connecting the premise to the conclusion using a series of logical steps. In other words, a direct proof involves using previously accepted facts or axioms to show that a statement or theorem is true.

Here is an example of a direct proof:

Statement: If x and y are odd integers, then xy is odd.

Proof:
Assume x and y are odd integers. This means that there exist integers a and b such that:
x = 2a + 1
y = 2b + 1
Then,
xy = (2a + 1)(2b + 1)
Expanding this expression, we get:

xy = 4ab + 2a + 2b + 1

We can see that all terms on the right side of the equation are even except for 1, which is an odd integer. Therefore, xy is odd.

Therefore, we have shown that if x and y are odd integers, then xy is odd by using a direct proof.

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