10-327/Homework Assignment 6 - Revision history

Anne.d: EDIT: moved Kai's HW6 solutions to new page

2010-12-11T02:55:41Z

EDIT: moved Kai's HW6 solutions to new page

← Older revision		Revision as of 22:55, 10 December 2010
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	* Yes. [[User:Drorbn\|Drorbn]] 06:47, 18 November 2010 (EST)		* Yes. [[User:Drorbn\|Drorbn]] 06:47, 18 November 2010 (EST)

	===Solution===
	Here is the solution to HW6 -Kai
	[http://katlas.math.toronto.edu/drorbn/images/2/2f/10-327a601.JPG page1]
	[http://katlas.math.toronto.edu/drorbn/images/4/46/10-327a602.JPG page2]
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	[http://katlas.math.toronto.edu/drorbn/images/e/ea/10-327a610.JPG page10]
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Xwbdsb: /* Solution */

2010-12-04T18:13:22Z

Solution

Xwbdsb: /* Solution */

2010-12-04T18:07:14Z

Solution

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	===Solution===		===Solution===
			Here is the solution to HW6 -Kai
	[http://katlas.math.toronto.edu/drorbn/images/7/77/10-327a501.JPG page1]		[http://katlas.math.toronto.edu/drorbn/images/7/77/10-327a501.JPG page1]
	[http://katlas.math.toronto.edu/drorbn/images/7/7e/10-327a502.JPG page2]		[http://katlas.math.toronto.edu/drorbn/images/7/7e/10-327a502.JPG page2]

Xwbdsb: /* Due date */

2010-12-04T18:06:58Z

Due date

← Older revision		Revision as of 14:06, 4 December 2010
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	* Yes. [[User:Drorbn\|Drorbn]] 06:47, 18 November 2010 (EST)		* Yes. [[User:Drorbn\|Drorbn]] 06:47, 18 November 2010 (EST)

			===Solution===
			[http://katlas.math.toronto.edu/drorbn/images/7/77/10-327a501.JPG page1]
			[http://katlas.math.toronto.edu/drorbn/images/7/7e/10-327a502.JPG page2]
			[http://katlas.math.toronto.edu/drorbn/images/6/6b/10-327a503.JPG page3]
			[http://katlas.math.toronto.edu/drorbn/images/6/61/10-327a504.JPG page4]
			[http://katlas.math.toronto.edu/drorbn/images/1/1d/10-327a505.JPG page5]
			[http://katlas.math.toronto.edu/drorbn/images/c/c6/10-327a506.JPG page6]
			[http://katlas.math.toronto.edu/drorbn/images/7/77/10-327a507.JPG page7]
			[http://katlas.math.toronto.edu/drorbn/images/d/df/10-327a508.JPG page8]
			[http://katlas.math.toronto.edu/drorbn/images/9/9d/10-327a509.JPG page9]
			[http://katlas.math.toronto.edu/drorbn/images/8/8e/10-327a510.JPG page10]
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			[http://katlas.math.toronto.edu/drorbn/images/7/70/10-327a512.JPG page12]
			[http://katlas.math.toronto.edu/drorbn/images/d/dc/10-327a513.JPG page13]
			[http://katlas.math.toronto.edu/drorbn/images/4/4d/10-327a514.JPG page14]
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Drorbn at 11:47, 18 November 2010

2010-11-18T11:47:06Z

← Older revision		Revision as of 07:47, 18 November 2010
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	Thank you Ian. What I meant to say was that if b: N -> [a,b] then we don't know for sure that b_tilda: beta_N -> [a,b] (ie the range of b_tilda could be [a-10, b+10] or something like this which is not contained in [a,b].) Thus, b_tilda is not necessarily an extension of b: N-> [a,b] but only an extension of b: N -> R. Do we need to take this into account for this question? Thanks.		Thank you Ian. What I meant to say was that if b: N -> [a,b] then we don't know for sure that b_tilda: beta_N -> [a,b] (ie the range of b_tilda could be [a-10, b+10] or something like this which is not contained in [a,b].) Thus, b_tilda is not necessarily an extension of b: N-> [a,b] but only an extension of b: N -> R. Do we need to take this into account for this question? Thanks.


	Now I think that the range of b_tilda must be inside [a,b] since b_tilda(beta_N) = b_tilda(N_closure) which is contained in b_tilda(N)_closure = b(N)_closure which is contained in [a,b]_closure = [a,b]. Is this correct? Thanks.		Now I think that the range of b_tilda must be inside [a,b] since b_tilda(beta_N) = b_tilda(N_closure) which is contained in b_tilda(N)_closure = b(N)_closure which is contained in [a,b]_closure = [a,b]. Is this correct? Thanks.

			* Yes. [[User:Drorbn\|Drorbn]] 06:47, 18 November 2010 (EST)

DavidB: /* Due date */

2010-11-18T04:55:01Z

Due date

← Older revision		Revision as of 00:55, 18 November 2010
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	Thank you Ian. What I meant to say was that if b: N -> [a,b] then we don't know for sure that b_tilda: beta_N -> [a,b] (ie the range of b_tilda could be [a-10, b+10] or something like this which is not contained in [a,b].) Thus, b_tilda is not necessarily an extension of b: N-> [a,b] but only an extension of b: N -> R. Do we need to take this into account for this question? Thanks.		Thank you Ian. What I meant to say was that if b: N -> [a,b] then we don't know for sure that b_tilda: beta_N -> [a,b] (ie the range of b_tilda could be [a-10, b+10] or something like this which is not contained in [a,b].) Thus, b_tilda is not necessarily an extension of b: N-> [a,b] but only an extension of b: N -> R. Do we need to take this into account for this question? Thanks.


			Now I think that the range of b_tilda must be inside [a,b] since b_tilda(beta_N) = b_tilda(N_closure) which is contained in b_tilda(N)_closure = b(N)_closure which is contained in [a,b]_closure = [a,b]. Is this correct? Thanks.

DavidB: /* Due date */

2010-11-18T04:12:28Z

Due date

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	<math>\tilde b </math> has to be bounded because <math>\beta{\mathbb N}</math> is compact. [[User:Ian\|Ian]] 18:54, 17 November 2010 (EST)		<math>\tilde b </math> has to be bounded because <math>\beta{\mathbb N}</math> is compact. [[User:Ian\|Ian]] 18:54, 17 November 2010 (EST)

			Thank you Ian. What I meant to say was that if b: N -> [a,b] then we don't know for sure that b_tilda: beta_N -> [a,b] (ie the range of b_tilda could be [a-10, b+10] or something like this which is not contained in [a,b].) Thus, b_tilda is not necessarily an extension of b: N-> [a,b] but only an extension of b: N -> R. Do we need to take this into account for this question? Thanks.

Ian: /* Due date */

2010-11-17T23:54:40Z

Due date

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	If F if the extension of the function f to (beta)N, then do we need to ensure that if mu is in (beta)N \ N, then F(mu) is in [0,1]^A ? (This is similar to how in class we extended a bounded function b to b_tilda but then we did not guarantee that b_tilda was bounded so then the range for b_tilda can be R but may not be any closed interval in R.) Thank you.		If F if the extension of the function f to (beta)N, then do we need to ensure that if mu is in (beta)N \ N, then F(mu) is in [0,1]^A ? (This is similar to how in class we extended a bounded function b to b_tilda but then we did not guarantee that b_tilda was bounded so then the range for b_tilda can be R but may not be any closed interval in R.) Thank you.

			<math>\tilde b </math> has to be bounded because <math>\beta{\mathbb N}</math> is compact. [[User:Ian\|Ian]] 18:54, 17 November 2010 (EST)

DavidB: /* Due date */

2010-11-17T23:00:34Z

Due date

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	* Yes and yes. [[User:Drorbn\|Drorbn]] 06:59, 17 November 2010 (EST)		* Yes and yes. [[User:Drorbn\|Drorbn]] 06:59, 17 November 2010 (EST)

			Question About Problem 3

			If F if the extension of the function f to (beta)N, then do we need to ensure that if mu is in (beta)N \ N, then F(mu) is in [0,1]^A ? (This is similar to how in class we extended a bounded function b to b_tilda but then we did not guarantee that b_tilda was bounded so then the range for b_tilda can be R but may not be any closed interval in R.) Thank you.

Drorbn at 11:59, 17 November 2010

2010-11-17T11:59:03Z

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	**Does the set of all bounded sequences has the same cardinality as the reals?[[User:Xwbdsb\|Xwbdsb]] 15:00, 16 November 2010 (EST)		**Does the set of all bounded sequences has the same cardinality as the reals?[[User:Xwbdsb\|Xwbdsb]] 15:00, 16 November 2010 (EST)

			----
	Question About Problem 2		Question About Problem 2

	Should we assume that {0,1} has the discrete topology and that {0,1}^N has the corresponding product topology and should we also assume that C has the subspace topology that it inherits from R? Thank you.		Should we assume that {0,1} has the discrete topology and that {0,1}^N has the corresponding product topology and should we also assume that C has the subspace topology that it inherits from R? Thank you.

			* Yes and yes. [[User:Drorbn\|Drorbn]] 06:59, 17 November 2010 (EST)

← Older revision		Revision as of 14:13, 4 December 2010
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	===Solution===		===Solution===
	Here is the solution to HW6 -Kai		Here is the solution to HW6 -Kai
	[http://katlas.math.toronto.edu/drorbn/images/7/77/10-~~327a501~~.JPG page1]		[http://katlas.math.toronto.edu/drorbn/images/2/2f/10-327a601.JPG page1]
	[http://katlas.math.toronto.edu/drorbn/images/7/7e/10-~~327a502~~.JPG page2]		[http://katlas.math.toronto.edu/drorbn/images/4/46/10-327a602.JPG page2]
	[http://katlas.math.toronto.edu/drorbn/images/6/6b/10-~~327a503~~.JPG page3]		[http://katlas.math.toronto.edu/drorbn/images/a/af/10-327a603.JPG page3]
	[http://katlas.math.toronto.edu/drorbn/images/6/61/10-~~327a504~~.JPG page4]		[http://katlas.math.toronto.edu/drorbn/images/9/9f/10-327a604.JPG page4]
	[http://katlas.math.toronto.edu/drorbn/images/1/1d/10-~~327a505~~.JPG page5]		[http://katlas.math.toronto.edu/drorbn/images/f/fa/10-327a605.JPG page5]
	[http://katlas.math.toronto.edu/drorbn/images/c/c6/10-~~327a506~~.JPG page6]		[http://katlas.math.toronto.edu/drorbn/images/b/b1/10-327a606.JPG page6]
	[http://katlas.math.toronto.edu/drorbn/images/7/77/10-~~327a507~~.JPG page7]		[http://katlas.math.toronto.edu/drorbn/images/1/1f/10-327a607.JPG page7]
	[http://katlas.math.toronto.edu/drorbn/images/d/df/10-~~327a508~~.JPG page8]		[http://katlas.math.toronto.edu/drorbn/images/d/d2/10-327a608.JPG page8]
	[http://katlas.math.toronto.edu/drorbn/images/9/9d/10-~~327a509~~.JPG page9]		[http://katlas.math.toronto.edu/drorbn/images/f/fe/10-327a609.JPG page9]
	[http://katlas.math.toronto.edu/drorbn/images/8/8e/10-~~327a510~~.JPG page10]		[http://katlas.math.toronto.edu/drorbn/images/e/ea/10-327a610.JPG page10]
	[http://katlas.math.toronto.edu/drorbn/images/7/75/10-~~327a511~~.JPG page11]		[http://katlas.math.toronto.edu/drorbn/images/5/54/10-327a611.JPG page11]
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