Can someone please tell me what the contrapositive to this statement is: the square of an even number is an even number.

I thought it was if a number is not even, then its square is not even.

Printable View

- Jul 1st 2008, 11:38 AMsjenkinsContrapositive
Can someone please tell me what the contrapositive to this statement is: the square of an even number is an even number.

I thought it was if a number is not even, then its square is not even. - Jul 1st 2008, 11:50 AMMoo
mmmm

Let

The first sentence is with some different syntax, but though the same meaning. I'll try to explain it... Grammatically, the "square of" comes__after__the "even number", in the sense that you have in a first time "an even number" and then, you get its "square". This is why you consider the square part as a consequence...

The contraposive is , that is to say :

.

In other words : "If the square of a number is not even, then this number is not even".

Is it clear ? Does it look correct to you ? (Worried) - Jul 1st 2008, 12:00 PMsjenkins
Yes, thanks so much Moo! I appreciate your constant willingness to help me!!!

- Jul 1st 2008, 01:16 PMSoroban
Hello, sjenkins!

Quote:

Can someone please tell me what the contrapositive to this statement is:

The square of an even number is an even number.

"The square of an even number is an even number."

We want: .

In other words: If the square of a number is odd, then the number is odd.

Too fast for me, Moo!

. - Jul 1st 2008, 01:21 PMsjenkinsMore contrapositive help
I just don't think I was thinking it through. Now, can anyone tell me if the following is a proof of this statement???

Assume that x is not an even number

Then x is not a multiple of 2

In other words, x=2k+1 for some integer k

So x^2=(2k+1)^2=4k^2+1=2(2k^2)+1=2n where n=2k^2+1

We check that n is also an integer which it is since k is an integer and the integers are closed under multiplication

So, x is not a multiple of 2

Therefore, x^2 is not even - Jul 1st 2008, 01:25 PMMoo
- Jul 1st 2008, 01:33 PMsjenkins
Why do I have such a difficult time understanding all of this. I am really trying here and nothing I do seems to work out.

- Jul 1st 2008, 03:42 PMalgebraic topology
- Jul 1st 2008, 03:50 PMPlato
- Jul 1st 2008, 04:17 PMalgebraic topology
I beg Moo’s pardon, the bar didn’t show up clearly on my browser (and it’s also the first time I’ve ever seen this notation being used.)

- Jul 1st 2008, 05:11 PMsjenkinsTrying this proof again...
Here's try number 2, does this work to prove if the square of a number is not even, then the number is not even.

Assume that x^2 is not an even number

Then x^2 is not a multiple of 2

In other words, x^2=(2k+1)^2 for some integer k

It follows that x=2k+1

Therefore x is not even - Jul 1st 2008, 05:34 PMReckoner
Your proof is okay, except for one little thing:

This does not work. Suppose . We can write , but . In other words, when taking the square root of a square, you have to be aware that . Of course, this is easy to rectify.

However, I think a direct proof of the original statement would be a lot simpler: Let be an even integer, i.e., , and then show, through a chain of implications, that must also be even. - Jul 1st 2008, 05:37 PMsjenkins
I agree that a direct proof of the original statement may have been easier, unfortunately the book said I had to create an indirect proof (Doh)

- Jul 1st 2008, 05:39 PMsjenkins
Can I ask what you would write to fix this problem...I'm confused!

- Jul 1st 2008, 05:44 PMReckoner