Proofs:Direct

The direct proof is relatively simple — by logically applying previous knowledge, we directly prove what is required.

Example 1

Prove that the sum of any two even integers Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} and $y$ is even.

Solution 1

We know that since $x$ and $y$ are even, they must have 2 as a factor. Then, we can write the following:

Let $x=2a$ , $y=2b$ , for some integers $a,b$

Then:

${\begin{matrix}x+y&=&2a+2b\\&=&2(a+b)\end{matrix}}$

by the distributive property of integers

The number $2(a+b)$ clearly has 2 as a factor, which implies it is even. Therefore, $x+y$ is even.

Example 2

Prove the following statement for non-zero integers $a,b,c$:

If $a$ divides $b$ and $b$ divides $c$ , then $a$ divides $c$ .

Solution 2

If an integer $x$ divides an integer $y$ , then we can write $y=qx$ , for some non-zero integer $q$ . So let's say that $b=qa$ and $c=rb$ , for some non-zero integers $q$ and $r$ . Then:

${\begin{matrix}c&=&rb\\&=&r(qa)\\&=&(rq)a\end{matrix}}$

by the associative property of integer multiplication.

But since $q$ and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} are integers, their product Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle qr} must also be an integer. Therefore, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} is the product of some integer multiplied by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} , so we get that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} divides Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} .

Resources

• Course Textbook, pages 53-57