# Simple Math Proofs

Alex Egg,

### Even and Odd Integers

An integer n is even if, and only if, n equals twice some integer. An integer n is odd if, and only if, n equals twice some integer plus 1.

Symbolically, if n is an integer, then

n is even $\Leftrightarrow \exists$ an integer k such that $n=2k$

n is odd $\Leftrightarrow \exists$ an integer k such that $n=2k+1$

### Prime Numbers

An integer $n$ is prime if, and only if, $n$>1 and for all positive integers $r$ and $s$, if $n=rs$, then either $r$ or $s$ equals $n$. An intger $n$ is composite if, and only if $n$>1 and $r=rs$ for some integers $r$ and $s$ with $1<r<n$ and $1<s<n$.
In symbols:

$n$ is prime $\Leftrightarrow \forall$ positive integers $r$ and $s$ if $n=rs$ then either $r=1$ and $s=n$ or $r=n$ and $s=1$.

$n$ is composite $\Leftrightarrow \exists$ positive integers $r$ and $s$ such that $n=rs$ and $1<r<n$ and $1<s<n$.

### The sum of any two even integers is even

Suppose $m$ and $n$ are even integers. By definition of even, $m=2r$ and $n=2s$ for some integers $r$ and $s$. Then

$m+n=2r+2s$

$=2(r+s)$

Let $t=r+s$. Note that $t$ is an integer because it is a sum of integers. hence $m+n=2t$ where t is an integer.

It follows by definition of even that $m+n$ is even. QED.

### Rational Number

A real number r is rational if, and only if, it can be expressed as a quotent of two integers with a nonzero denominator. A real number that is not rational is irrational.
More formally, if r is a real number, then

$r$ is rational $\Leftrightarrow \exists$ integers $a and b$ such that $r=\frac{a}{b} and b\neq0.$

### The sum of any two rational numbers is rational

Suppose r and s are rational numbers. Then by definition of rational, $r=\frac{a}{b} and \frac{c}{d}$ for some integers $a,b,c, and d$ with $b\neq0$ and $d\neq0$. Thus

$r+s=\frac{a}{b}+\frac{c}{d}$

$=\frac{ad+bc}{bd}$

Let $p=ad+bc$ and $q=bd$. Then p and q are integers because products and sums of integers are integers and because a,b,c and d are all integers. Also $q\neq0$ by the zero product property. Thus

$r+s=\frac{p}{q}$ where p and q are integers and $q\neq0$.

Therefore, $r+s$ is rational by definition of a rational number. QED.