Difference between revisions of "Proofs:Direct"
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| + | 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 <math>x</math> and <math>y</math> is even. | ||
| + | |||
| + | '''Solution 1''' | ||
| + | |||
| + | We know that since <math>x</math> and <math>y</math> are even, they must have 2 as a factor. Then, we can write the following: | ||
| + | :Let <math>x=2a</math> , <math>y=2b</math> , for some integers <math>a,b</math> | ||
| + | Then: | ||
| + | :<math>\begin{matrix}x+y&=&2a+2b\\&=&2(a+b)\end{matrix}</math> | ||
| + | by the distributive property of integers | ||
| + | |||
| + | The number <math>2(a+b)</math> clearly has 2 as a factor, which implies it is even. Therefore, <math>x+y</math> is even. | ||
| + | |||
| + | '''Example 2''' | ||
| + | |||
| + | Prove the following statement for non-zero integers <math>a,b,c</math>: | ||
| + | |||
| + | If <math>a</math> divides <math>b</math> and <math>b</math> divides <math>c</math> , then <math>a</math> divides <math>c</math> . | ||
| + | |||
| + | '''Solution 2''' | ||
| + | |||
| + | If an integer <math>x</math> divides an integer <math>y</math> , then we can write <math>y=qx</math> , for some non-zero integer <math>q</math> . So let's say that <math>b=qa</math> and <math>c=rb</math> , for some non-zero integers <math>q</math> and <math>r</math> . Then: | ||
| + | :<math>\begin{matrix}c&=&rb\\&=&r(qa)\\&=&(rq)a\end{matrix}</math> | ||
| + | by the associative property of integer multiplication. | ||
| + | |||
| + | But since <math>q</math> and <math>r</math> are integers, their product <math>qr</math> must also be an integer. Therefore, <math>c</math> is the product of some integer multiplied by <math>a</math> , so we get that <math>a</math> divides <math>c</math> . | ||
| + | |||
==Resources== | ==Resources== | ||
* [https://link-springer-com.libweb.lib.utsa.edu/content/pdf/10.1007%2F978-1-4419-7127-2.pdf Course Textbook], pages 53-57 | * [https://link-springer-com.libweb.lib.utsa.edu/content/pdf/10.1007%2F978-1-4419-7127-2.pdf Course Textbook], pages 53-57 | ||
Latest revision as of 10:47, 1 October 2021
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 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 y} is even.
Solution 1
We know that since 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 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 y} are even, they must have 2 as a factor. Then, we can write the following:
- Let 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=2a} , 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 y=2b} , for some 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 a,b}
Then:
by the distributive property of integers
The number 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 2(a+b)} clearly has 2 as a factor, which implies it is even. 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 x+y} is even.
Example 2
Prove the following statement for non-zero 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 a,b,c} :
If 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 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 b} 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} , then 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} .
Solution 2
If an integer 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} divides an integer , then we can write 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 y=qx} , for some non-zero integer 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 q} . So let's say 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 b=qa} 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 c=rb} , for some non-zero integers 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} . Then:
by the associative property of integer multiplication.
But since 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 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, 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 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
