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Re: I've having an argument with my maths teacher

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sdfgw is not online. sdfgw
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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?

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pighead10 is not online. pighead10
Joined: 03 May 2009
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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
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sdfgw is not online. sdfgw
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14 Jan 2012 02:27 PM
ok bad wording

n^3 + 1 is an odd integer

THEREFORE

n is an even integer
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pighead10 is not online. pighead10
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14 Jan 2012 02:29 PM
then I agree with you, seeing as the statement is obviously not true if you can prove it
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sncplay42 is not online. sncplay42
Joined: 27 Nov 2008
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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.
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blocco is not online. blocco
Joined: 14 Aug 2008
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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.
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sdfgw is not online. sdfgw
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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 <->
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TheMyrco is not online. 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



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sdfgw is not online. sdfgw
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14 Jan 2012 02:32 PM
@snc & blocco:

yay
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sncplay42 is not online. sncplay42
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14 Jan 2012 02:32 PM
"Given n = 1.
n^3 + 1 = 4"

"n is even"
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sdfgw is not online. sdfgw
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14 Jan 2012 02:37 PM
"Given n = 1."

why would anyone give you that
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crazytomato4 is not online. crazytomato4
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14 Jan 2012 02:38 PM
@myrco:

1 + 1 = 4

i dont get it
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sdfgw is not online. sdfgw
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14 Jan 2012 02:38 PM
^

didn't notice that

AHA what
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Varp is not online. Varp
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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.
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Oysi is not online. Oysi
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14 Jan 2012 02:42 PM
[ Content Deleted ]
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Oysi is not online. Oysi
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14 Jan 2012 02:43 PM
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sdfgw is not online. sdfgw
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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
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stravant is not online. stravant
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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.
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Oysi is not online. Oysi
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14 Jan 2012 02:44 PM
[ Content Deleted ]
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pighead10 is not online. pighead10
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14 Jan 2012 02:55 PM
I was right

what
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sdfgw is not online. sdfgw
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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
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blocco is not online. blocco
Joined: 14 Aug 2008
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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
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pighead10 is not online. pighead10
Joined: 03 May 2009
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14 Jan 2012 03:00 PM
"n ∊ ℕ"
I don't recognise the last 2 symbols, what do they mean
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blocco is not online. blocco
Joined: 14 Aug 2008
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14 Jan 2012 03:00 PM
"n ∊ ℕ"
means 'n' is an element of 'ℕ', or all the naturals
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pighead10 is not online. pighead10
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14 Jan 2012 03:02 PM
" element of 'ℕ'"

lost me tehre
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