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#1 2008-02-20 06:06:21
- Daniel123
- Member
- Registered: 2007-05-23
- Posts: 663
Prove inequality
Hello.
I was wondering if this is a valid proof?
Prove that (a+b)² ≤ 2a² + 2b²
Suppose that the statement is not true i.e.
(a+b)² ≥ 2a² + 2b²
a² + 2ab + b² ≥ 2a² + 2b²
a² + b² ≤ 2ab
a² - 2ab + b² ≤ 0
(a-b)² ≤ 0
This is an absurdity, and therefore the original statement must be true.
Thanks.
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#2 2008-02-20 06:17:28
- LuisRodg
- Real Member
- Registered: 2007-10-23
- Posts: 322
Re: Prove inequality
What if a = b = 0?
Do you know what values a and b can take?
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#3 2008-02-20 06:23:25
- Daniel123
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- Registered: 2007-05-23
- Posts: 663
Re: Prove inequality
Nope, there is no other information.
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#4 2008-02-20 06:29:30
- LuisRodg
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- Registered: 2007-10-23
- Posts: 322
Re: Prove inequality
Then that does not prove it unless you state that a and b cannot be equal.
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#5 2008-02-20 06:37:45
- JaneFairfax
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- Registered: 2007-02-23
- Posts: 6,868
Re: Prove inequality
The negation of
is , NOT .Be careful.
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#6 2008-02-20 07:08:44
- Daniel123
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- Registered: 2007-05-23
- Posts: 663
Re: Prove inequality
Hmm.. how would you do it then?
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#7 2008-02-20 07:12:36
- mathsyperson
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- Registered: 2005-06-22
- Posts: 4,900
Re: Prove inequality
The same way you did it. What Jane is saying is that you should start off by assuming that (a+b)² > 2a² + 2b².
Then following the same steps, you end up with (a-b)²<0.
Which actually is an absurdity.
Why did the vector cross the road?
It wanted to be normal.
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#8 2008-02-20 07:20:07
- Daniel123
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- Registered: 2007-05-23
- Posts: 663
Re: Prove inequality
Ohh right ok. I've never done a proof by contradiction before. 
Thanks.
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#9 2008-02-20 09:11:01
- JaneFairfax
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- Registered: 2007-02-23
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Re: Prove inequality
Alternatively, you can start with the inequality
(which is always true) and work your way to the given inequality, thereby avoiding proof by contradiction. 
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#10 2008-02-20 09:13:45
- Daniel123
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- Registered: 2007-05-23
- Posts: 663
Re: Prove inequality
I thought of that after I had finished - but unless you knew that if the statement wasn't true it would result in in (a-b)² ≤ 0 then it would be quite hard to think of.
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#11 2008-02-20 09:16:15
- Ricky
- Moderator
- Registered: 2005-12-04
- Posts: 3,791
Re: Prove inequality
This is why most inequality proofs are presented backwards from the way they are derived. This is pretty much standard.
"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."
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