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kno how to go about proving this?

for any functionals f and g.
//or at least kno of an example where equality doesnt hold? =/
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boblion   Andorra. Apr 09 2009 17:15. Posts 354 | | |
f(t) + g(t) =< sup f(t) + sup g(t)
sup ( f(t) + g(t) ) =< sup ( sup f(t) + sup g(t) ) = sup f(t) + sup g(t)
no ?
i'm rusty :< |
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I worship variance but she is a bitch. | Last edit: 09/04/2009 17:15 |
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spencer   United States. Apr 09 2009 17:18. Posts 23 | | |
Say S = sup f(t)+g(t), F = sup f(t), G = sup g(t). Note that F > f(t), G > g(t) for all t in A so that F+G is an upper bound for f+g, but if S > F+G, the property of the supremum implies F+G is not an upper bound for f+g. |
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spencer   United States. Apr 09 2009 17:22. Posts 23 | | |
the key idea is that supremum is the least upper bound, so once you find an upper bound, you know the supremum has to be <= to it. thats basically what boblion did too. |
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heh..that was easy -_-...ya im rusty too. any examples of functions where equality doesnt hold? equality should hold for all continuous functions i think |
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spencer   United States. Apr 09 2009 17:29. Posts 23 | | |
im pretty sure it's true for all real functions. the proof assumes no special properties about f or g.
edit: misread your question, so ignore this |
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k3rn3l   Hungary. Apr 09 2009 17:31. Posts 217 | | |
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boblion   Andorra. Apr 09 2009 17:31. Posts 354 | | |
| On April 09 2009 16:24 killThemDonks wrote:
heh..that was easy -_-...ya im rusty too. any examples of functions where equality doesnt hold?
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No idea but i guess it is always true.
| On April 09 2009 16:24 killThemDonks wrote:
equality should hold for all continuous functions i think |
For sure. |
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I worship variance but she is a bitch. | |
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are any of u in pure math? i may have some tougher questions..not homework or anything...just a refresher |
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boblion   Andorra. Apr 09 2009 17:38. Posts 354 | | |
| On April 09 2009 16:35 killThemDonks wrote:
are any of u in pure math? i may have some tougher questions..not homework or anything...just a refresher |
I'm math/eco but i'm bad and rusty  |
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I worship variance but she is a bitch. | |
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spencer   United States. Apr 09 2009 17:41. Posts 23 | | |
yeah I am, just finished a class in real analysis last semester actually |
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hmm wait....equality holds for continuous functions in general....or continuous functions over a closed set...?? bah...soo rusty |
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Zalfor   United States. Apr 09 2009 18:03. Posts 2236 | | |
i'm a math major.
equality holds |
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no wait...im an idiot...brb |
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now im confused...cuz say f, and g are continuous over some (closed) set T,
=> sup{ f(t): t in T} = max{ f(t): t in T} ...same holds for g(t)
Let h(t) = f(t) + g(t), it follows that h(t) is also continuous over T ==> sup h(t) = max h(t) = max( f(t) + g(t) ) =/= max( f(t) ) + max ( g(t) )
:S.. amirite?
take functions N(0,1) and N(1,1) for example over T = [-2,2]
so equality doesnt hold for cts functions????
math is idiotic..... |
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lazymej   Canada. Apr 09 2009 18:21. Posts 2897 | | |
lol math  |
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| On April 09 2009 17:14 killThemDonks wrote:
now im confused...cuz say f, and g are continuous over some (closed) set T,
=> sup{ f(t): t in T} = max{ f(t): t in T} ...same holds for g(t)
Let h(t) = f(t) + g(t), it follows that h(t) is also continuous over T ==> sup h(t) = max h(t) = max( f(t) + g(t) ) =/= max( f(t) ) + max ( g(t) )
:S.. amirite?
take functions N(0,1) and N(1,1) for example over T = [-2,2]
so equality doesnt hold for cts functions????
math is idiotic..... |
halp! |
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spencer   United States. Apr 09 2009 18:52. Posts 23 | | |
yeah, equality definitely doesn't hold even for continuous functions in general - it's true only if the maxes of f and g coincide at the same point |
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Day[9]   United States. Apr 09 2009 18:56. Posts 3447 | | |
here's an easy one
f(t) = cos t
g(t) = -cos t
individually, each one oscillates between 1 and -1, thus each has a supremum of 1
add them together, and you get f(t) + g(t) = 0. the supremum is 0
zing |
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Day[9]   United States. Apr 09 2009 18:57. Posts 3447 | | |
that is equality doesn't hold for those bad boys |
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Day[9]   United States. Apr 09 2009 18:58. Posts 3447 | | |
that is sup ( f(t) + g(t) ) = sup ( 0 ) = 0 < sup f(t) + sup g(t) = 1+1 = 2
0 < 2
eh? |
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NiTE   Croatia. Apr 09 2009 19:12. Posts 366 | | |
| On April 09 2009 16:31 k3rn3l wrote:
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I lol'd |
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Have an opinion about what I said? please say it | |
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^ equality doesnt hold necessarily for continuous functions (not sure why i thought so in the first place :S). |
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boblion   Andorra. Apr 09 2009 20:23. Posts 354 | | |
omg
i'm retarded. I thought Day said that the whole "=<" was wrong lol |
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I worship variance but she is a bitch. | |
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Bigbobm   United States. Apr 15 2009 04:38. Posts 5512 | | |
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Its time to stop thinking like a bitch and think smart like a poker player - ket | |
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rockman255   Canada. Apr 16 2009 01:07. Posts 4471 | | |
how did i receive a degree in mathematics O___o |
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rockman255: its not easy being superman U N0 MySteeZ: mega man. rockman255: same thing U N0 MySteeZ: no | |
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xx frankie   United States. Apr 16 2009 11:34. Posts 57 | | |
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PokerDoc88   Australia. Apr 16 2009 13:37. Posts 3527 | | |
| On April 09 2009 18:12 NiTE wrote:
I lol'd
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LMFAO this is the first thing I thought when I saw the equation. Well, not exactly a cat saying "sup" but definitely the "Sup?" mentality came to mind.
Wish I could help, I haven't touched any math for 3 years and this looks beyond anything I knew even when I was doing this back in high school. I miss math... |
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