01204211/activity2 logic and proofs - ประวัติรุ่นแก้ไข

Jittat: /* Homework 2 */

2015-09-09T04:19:19Z

Homework 2

2015-08-27T05:19:54Z

B. Proofs

2015-08-27T04:39:44Z

B. Proofs

2015-08-27T01:43:00Z

Homework 2

2015-08-27T01:41:44Z

Homework 2

2015-08-27T01:39:18Z

Homework 2

2015-08-27T01:35:52Z

Homework 2

2015-08-27T01:34:30Z

B. Proofs

2015-08-27T01:31:39Z

Homework 2

2015-08-27T01:31:29Z

Homework 2

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 == Homework 2 ==
-: ''Due date: 9 Sep 2015''
+: ''Due date: 11 Sept. 2015 <del>9 Sep 2015</del>''
 '''H1.''' Prove the following statement.
@@ แถว 94: / แถว 94: @@
 '''H4.''' Prove that for any positive integer <math>n</math>, <math>n</math> is an odd number if and only if <math>5n+6</math> is odd.
 '''Sources:'''
 * H2, H3, and H4 is from Rosen, Examples 33, 23, and 39, Ch.1, pp. 67.

←รุ่นแก้ไขก่อนหน้า		รุ่นแก้ไขเมื่อ 05:19, 27 สิงหาคม 2558
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	'''B2.''' If <math>x</math> and <math>y</math> are integers and <math>x^2 + y^2</math> are even, then <math>x+y</math> is even.		'''B2.''' If <math>x</math> and <math>y</math> are integers and <math>x^2 + y^2</math> are even, then <math>x+y</math> is even.
		+
		+	: ''Hints:'' When you want to prove this statement: "If x and y are integers and <math>x^2 + y^2</math> are even, then x + y is even. ". You can think of it as: "If x and y are integers, then (if <math>x^2 + y^2</math>, then x+y is even)". That is because (P and Q) => R is equivalent to (P => (Q => R)). Therefore, in this case, you can start by assuming that x and y are integers.

←รุ่นแก้ไขก่อนหน้า		รุ่นแก้ไขเมื่อ 04:39, 27 สิงหาคม 2558
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−	'''B4.''' If <math>x</math> is a non-zero rational number ~~and~~ <math>y</math> is irrational, then <math>xy</math> is irrational.	+	'''B4.''' Assume that <math>x</math> is a non-zero rational number. Prove that if <math>y</math> is irrational, then <math>xy</math> is irrational.

@@ แถว 92: / แถว 92: @@
 '''H4.''' Prove that for any positive integer <math>n</math>, <math>n</math> is an odd number if and only if <math>5n+6</math> is odd.
 '''Sources:'''
 * H2, H3, and H4 is from Rosen, Examples 33, 23, and 39, Ch.1, pp. 67.

@@ แถว 89: / แถว 89: @@
 '''H3.''' Prove that <math>|xy|=|x||y|</math>, where <math>x</math> and <math>y</math> are real numbers.
 '''Sources:'''
-* H2 and H3 is from Rosen, Examples 33 and 23, Ch.1, pp. 67.
+* H2, H3, and H4 is from Rosen, Examples 33, 23, and 39, Ch.1, pp. 67.

@@ แถว 85: / แถว 85: @@
-'''H2.''' Prove that <math>|xy|=|x||y|</math>, where <math>x</math> and <math>y</math> are real numbers.
+'''H2.''' Prove that if <math>x</math> and <math>y</math> are real numbers, then <math>\max(x,y) + \min(x,y) = x+y</math>
 '''Sources:'''
-* H2 is from Rosen, Example 23, Ch.1, pp. 67.
+* H2 and H3 is from Rosen, Examples 33 and 23, Ch.1, pp. 67.

@@ แถว 44: / แถว 44: @@
-'''B4.'''
+'''B4.''' If <math>x</math> is a non-zero rational number and <math>y</math> is irrational, then <math>xy</math> is irrational.
-'''Source:''' B1 and B3 are from ''เอกสารประกอบการสอนวิชา Math 217 แนวคิดหลักมูลของคณิตศาสตร์ โดย รุ่งนภา ภักดีสู่สุข ม.เชียงใหม่''.  B2 is from [https://www.whitman.edu/mathematics/higher_math_online/section02.01.html#exercises whitman.edu].
+'''Source:'''
 === C. Proofs by contradiction ===

←รุ่นแก้ไขก่อนหน้า		รุ่นแก้ไขเมื่อ 01:31, 27 สิงหาคม 2558
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	: If integer ''c'' divides both integers ''a'' and ''b'', then ''c'' divides ''a - b''.		: If integer ''c'' divides both integers ''a'' and ''b'', then ''c'' divides ''a - b''.
		+
		+
		+	'''H2.''' Prove that <math>\|xy\|=\|x\|\|y\|/math>, where <math>x</math> and <math>y</math> are real numbers.
		+
		+
		+	'''Sources:'''
		+	* H2 is from Rosen, Example 23, Ch.1, pp. 67.