Lila at 16:47, 1 October 2021

2021-10-01T16:47:25Z

Lila: Created page with "==Resources== * [https://link-springer-com.libweb.lib.utsa.edu/content/pdf/10.1007%2F978-1-4419-7127-2.pdf Course Textbook], pages 53-57"

2021-09-22T16:07:42Z

Created page with "==Resources== * [https://link-springer-com.libweb.lib.utsa.edu/content/pdf/10.1007%2F978-1-4419-7127-2.pdf Course Textbook], pages 53-57"

New page

==Resources==
* [https://link-springer-com.libweb.lib.utsa.edu/content/pdf/10.1007%2F978-1-4419-7127-2.pdf Course Textbook], pages 53-57

← Older revision		Revision as of 16:47, 1 October 2021
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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

Proofs:Direct - Revision history

Lila at 16:47, 1 October 2021

Lila: Created page with "==Resources== * [https://link-springer-com.libweb.lib.utsa.edu/content/pdf/10.1007%2F978-1-4419-7127-2.pdf Course Textbook], pages 53-57"