sdfgw
|
  |
 |
| Joined: 08 Jan 2009 |
| Total Posts: 41681 |
|
|
| 14 Jan 2012 02:24 PM |
The statement:
n^3 + 1 is an odd integer implies that n is an even integer.
This came in C1 (first unit) A-level maths paper from OCR if it means anything to you:
her answer: yes, since is n is even then n^3 is even, hence n^3 + 1 is odd
my answer: no as you can disprove it with a counterexample, something like n^3 + 1 = 11, n = root3(10) which is not an integer
she says that my answer is looking too far into it
Anyone been through this know the likely answer?
|
|
|
| Report Abuse |
|
|
pighead10
|
  |
| Joined: 03 May 2009 |
| Total Posts: 10341 |
|
|
| 14 Jan 2012 02:26 PM |
Since the question says 'implies' I would agree with her, and also agree that providing a counter example for something that only implies something *is* looking too far into it
but then what do i know |
|
|
| Report Abuse |
|
|
sdfgw
|
  |
 |
| Joined: 08 Jan 2009 |
| Total Posts: 41681 |
|
|
| 14 Jan 2012 02:27 PM |
ok bad wording
n^3 + 1 is an odd integer
THEREFORE
n is an even integer |
|
|
| Report Abuse |
|
|
pighead10
|
  |
| Joined: 03 May 2009 |
| Total Posts: 10341 |
|
|
| 14 Jan 2012 02:29 PM |
| then I agree with you, seeing as the statement is obviously not true if you can prove it |
|
|
| Report Abuse |
|
|
sncplay42
|
  |
| Joined: 27 Nov 2008 |
| Total Posts: 11891 |
|
|
| 14 Jan 2012 02:30 PM |
You're right; the statement is the wrong way round.
It _would_ be true that n is even implies n^3+1 is odd, but the other way round isn't true because your counter example disproves it. |
|
|
| Report Abuse |
|
|
blocco
|
  |
| Joined: 14 Aug 2008 |
| Total Posts: 29474 |
|
|
| 14 Jan 2012 02:30 PM |
| Doesn't it only imply that n^3 is even, not n? I would assume that if the statement doesn't explicitly say that n^3 has an integer cube-root, that n doesn't have to be an integer. |
|
|
| Report Abuse |
|
|
sdfgw
|
  |
 |
| Joined: 08 Jan 2009 |
| Total Posts: 41681 |
|
|
| 14 Jan 2012 02:31 PM |
mmm
I suppose if we're going to be precise the EXACT question was to choose the correct connector symbol: <-, ->, or <-> to go inbetween
n^3 + 1 is an odd integer
and
n is an even integer
I chose <-, she chose <-> |
|
|
| Report Abuse |
|
|
TheMyrco
|
  |
| Joined: 13 Aug 2011 |
| Total Posts: 15105 |
|
|
| 14 Jan 2012 02:32 PM |
Not always.
Given n = 2. n^3 + 1 = 9
Given n = 1. n^3 + 1 = 4
|
|
|
| Report Abuse |
|
|
sdfgw
|
  |
 |
| Joined: 08 Jan 2009 |
| Total Posts: 41681 |
|
| |
|
sncplay42
|
  |
| Joined: 27 Nov 2008 |
| Total Posts: 11891 |
|
|
| 14 Jan 2012 02:32 PM |
"Given n = 1. n^3 + 1 = 4"
"n is even" |
|
|
| Report Abuse |
|
|
sdfgw
|
  |
 |
| Joined: 08 Jan 2009 |
| Total Posts: 41681 |
|
|
| 14 Jan 2012 02:37 PM |
"Given n = 1."
why would anyone give you that |
|
|
| Report Abuse |
|
|
|
| 14 Jan 2012 02:38 PM |
@myrco:
1 + 1 = 4
i dont get it |
|
|
| Report Abuse |
|
|
sdfgw
|
  |
 |
| Joined: 08 Jan 2009 |
| Total Posts: 41681 |
|
|
| 14 Jan 2012 02:38 PM |
^
didn't notice that
AHA what |
|
|
| Report Abuse |
|
|
Varp
|
  |
| Joined: 18 Nov 2009 |
| Total Posts: 5333 |
|
|
| 14 Jan 2012 02:41 PM |
| Your math teacher reached the wrong conclusion, although the logic she used is solid. She proved that n being even implies that n^3 + 1 is odd, but that's irrelevant since question asked if the converse is true. Your answer is correct, as it's a proof by contradiction that the converse must not be true. |
|
|
| Report Abuse |
|
|
Oysi
|
  |
| Joined: 06 Jul 2009 |
| Total Posts: 9058 |
|
| |
|
Oysi
|
  |
| Joined: 06 Jul 2009 |
| Total Posts: 9058 |
|
| |
|
sdfgw
|
  |
 |
| Joined: 08 Jan 2009 |
| Total Posts: 41681 |
|
|
| 14 Jan 2012 02:43 PM |
i'm beginning to feel slightly more hopeful about this
At least if we're ALL wrong then the grade boundries will probably go down, anyway |
|
|
| Report Abuse |
|
|
stravant
|
  |
 |
| Joined: 22 Oct 2007 |
| Total Posts: 2893 |
|
|
| 14 Jan 2012 02:44 PM |
Your error is in the notation. When you write: "n^3 + 1" you're only interested in a domain of all n ∊ ℕ. That means counterexample is irrelevant because the "n" that you found is not in the domain for which the claim is made (The claim only states it's that way for natural numbers, which yours is not). Once you recognize that, a proof becomes relatively easy by using the contrapositive (That is, prove that not Q implies not P rather than P implies Q, as those two statements are logically equivalent):
n^3 + 1 is an odd integer implies that n is an even integer. contrapositive: n is an odd integer implies that n^3 + 1 is an even integer. N is an odd integer means we can represent n as "2k-1" for some other natural number k. Then we re-write the initial identity: n^3 + 1 = (2k-1)^3 + 1 = 8k^3 -12k^2 +6k -1 + 1 = 2(4k^3 - 6k^2 + 3k) = n^3 + 1 Which is two-times some natural number, and obviously even, that is: n is odd implies n^3 + 1 is even. And by contrapositive: n^3 + 1 is odd implies that n is even. |
|
|
| Report Abuse |
|
|
Oysi
|
  |
| Joined: 06 Jul 2009 |
| Total Posts: 9058 |
|
| |
|
pighead10
|
  |
| Joined: 03 May 2009 |
| Total Posts: 10341 |
|
| |
|
sdfgw
|
  |
 |
| Joined: 08 Jan 2009 |
| Total Posts: 41681 |
|
|
| 14 Jan 2012 02:58 PM |
"you're only interested in a domain of all n ∊ ℕ."
now where did you get that
If the statement is
n^3 + 1 is an odd integer -> n is an even integer
nowhere does it say that 'n' is an integer, except in what we are trying to prove
and I think if I had to use a "contrapositive" I would've been taught that |
|
|
| Report Abuse |
|
|
blocco
|
  |
| Joined: 14 Aug 2008 |
| Total Posts: 29474 |
|
|
| 14 Jan 2012 02:59 PM |
Why does the word "contrapositive" sound like "against good" which is "bad"? Or am I taking a pessimistic approach |
|
|
| Report Abuse |
|
|
pighead10
|
  |
| Joined: 03 May 2009 |
| Total Posts: 10341 |
|
|
| 14 Jan 2012 03:00 PM |
"n ∊ ℕ" I don't recognise the last 2 symbols, what do they mean |
|
|
| Report Abuse |
|
|
blocco
|
  |
| Joined: 14 Aug 2008 |
| Total Posts: 29474 |
|
|
| 14 Jan 2012 03:00 PM |
"n ∊ ℕ" means 'n' is an element of 'ℕ', or all the naturals |
|
|
| Report Abuse |
|
|
pighead10
|
  |
| Joined: 03 May 2009 |
| Total Posts: 10341 |
|
|
| 14 Jan 2012 03:02 PM |
" element of 'ℕ'"
lost me tehre |
|
|
| Report Abuse |
|
|