In discrete math For all odd integers m and n mn is an even

In discrete math,

\"For all odd integers m and n, m+n is an even integer\"

Theorem: ___________________________________

Proof: _________________________________

Solution

Prove that \" x if P(x) then Q(x)

The meaning of the statement is: All x that have the property P, have also the property Q.

Method:

Explanation of step 1: How do we choose x arbitrary?

We use the definition of the property P to give a general representation of x

E.G.

If P is the property to be an even number, then x would be represented as 2n,
where n is any integer.

If P is the property to be an odd number, then x would be represented as 2n +1,
where n is any integer.

Explanation of step 2: How do we show that some property is true?

We use the definition of that property, and apply the rules of inference.

Example of using definitions and inference rules:

Prove that 10 is an even number.

Proof:

By definition an even number can be written as 2 * n
10 can be written as 2 * 5
Hence 10 is an even number.

The inference rule here is Modus ponens:

P