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 Post subject: Demonstrating if a number is evenPosted: Sat, 2 Apr 2011 22:05:06 UTC
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Joined: Sun, 20 Mar 2011 05:26:38 UTC
Posts: 25
I'm looking at a problem, and want to check if my attempt is right:
Problem
For n an integer number, consider the following definition:
Def. 1: is even

Prove that:
is even is even.

My attempt
is even
using Def. 1.

is even since

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 Post subject: Posted: Sat, 2 Apr 2011 22:19:15 UTC
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Joined: Wed, 1 Oct 2003 04:45:43 UTC
Posts: 9632
In your attempt, you have only proven the forward direction, which is "if n is even then n^2 is also even."

But now you have to do the backward direction, "if n^2 is even, then n is also even."
If n^2 = 2k for some integer k, then can you explain why n = 2p for some other integer p?

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 Post subject: Posted: Sat, 2 Apr 2011 22:39:32 UTC
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Joined: Sun, 20 Mar 2011 05:26:38 UTC
Posts: 25
Thanks for the help.

Ok, so here is my new attempt.
What I'm not sure about is if writing is enough to justify.

"":
If is even
using Def. 1

since

"":
If is even
using Def. 1
, because
is even

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 Post subject: Posted: Tue, 5 Apr 2011 08:26:35 UTC
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Joined: Sun, 20 Mar 2011 05:26:38 UTC
Posts: 25
I found another way of approaching this.

is even is even:
If is even

is even

(Using the contrapositive)
is not even is not even:
If is not even

is not even

is even is even.

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 Post subject: Posted: Tue, 5 Apr 2011 08:34:45 UTC
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I am not really convinced by the reverse direction.
How does imply ?

Instead, you might find it easier to prove: "if n is odd, then n^2 is odd."

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 Post subject: Posted: Tue, 5 Apr 2011 08:37:01 UTC
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Joined: Sun, 20 Mar 2011 05:26:38 UTC
Posts: 25
Ok, your point is convincing. Will do it that way. Thanks.

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 Post subject: Posted: Tue, 5 Apr 2011 08:54:40 UTC
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Joined: Sun, 20 Mar 2011 05:26:38 UTC
Posts: 25
is even is even:
If is even

is even

(Using the contrapositive)
is not even is not even is odd is odd.
If is odd

is odd

is even is even.

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 Post subject: Posted: Tue, 5 Apr 2011 18:56:07 UTC
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mathtinkerer wrote:

is odd.

How do you know that is an odd number?

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 Post subject: Posted: Tue, 5 Apr 2011 19:00:54 UTC
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Joined: Wed, 30 Mar 2005 04:25:14 UTC
Posts: 12097
Location: Austin, TX
Matt wrote:
mathtinkerer wrote:

is odd.

How do you know that is an odd number?

He doesn't, he hasn't yet decided to let

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 Post subject: Posted: Tue, 5 Apr 2011 22:04:54 UTC
 S.O.S. Oldtimer

Joined: Sat, 16 Aug 2008 04:47:19 UTC
Posts: 208
Matt wrote:
mathtinkerer wrote:

is odd.

How do you know that is an odd number?

He doesn't, he hasn't yet decided to let

perhaps you mean ?

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 Post subject: Posted: Tue, 5 Apr 2011 22:07:16 UTC
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Joined: Wed, 30 Mar 2005 04:25:14 UTC
Posts: 12097
Location: Austin, TX
daveyinaz wrote:
Matt wrote:
mathtinkerer wrote:

is odd.

How do you know that is an odd number?

He doesn't, he hasn't yet decided to let

perhaps you mean ?

Yes of course, didn't you read my mind to see what I was thinking?

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 Post subject: Posted: Wed, 6 Apr 2011 06:35:27 UTC
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Joined: Sun, 20 Mar 2011 05:26:38 UTC
Posts: 25
daveyinaz wrote:
Matt wrote:
mathtinkerer wrote:

is odd.

How do you know that is an odd number?

He doesn't, he hasn't yet decided to let

perhaps you mean ?

Yes, I still need to do that. Thanks everybody.

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