Discrete Math Proof: Proving Equivalence of 4 Statements

[MarcL]

Homework Statement

Prove that the following four statements are equivalent:
(a) n² is odd.
(b) 1 − n is even.
(c) n² is odd.
(d) n² + 1 is even.

Homework Equations

None really, just the use of different proofs ( indirect, etc...)

The Attempt at a Solution

I'm having trouble with this one because of (2) things. First, the question makes no sense ( unless I'm reading it wrong) because, as I understand it now, it is asking to prove all statements are equivalent BUT 2 of them are eve and the rest is odd. Secondly, I'm having trouble with it because of the definition that n=2k is EVEN and n=2k+1 is odd because I tried this:

Supposle n is odd.
n=2k therefore n=(2k)²= 2²k² = 2(2k²) which goes against the definition.

Can anyone point me in the right direction?

 

[phinds]

(a) n² is odd.
(b) 1 − n is even.
(c) n² is odd.
(d) n² + 1 is even.

Uh ... are you sure you copied (a) and (c) correctly?

 

[MarcL]

I promise you I did, if you want to see the assignment yourself you can check :P --> #4 http://users.encs.concordia.ca/~grahne/comp232/assgn2.pdf

 

[phinds]

I promise you I did, if you want to see the assignment yourself you can check :P --> #4 http://users.encs.concordia.ca/~grahne/comp232/assgn2.pdf

Exactly. You copied it incorrectly, as is obvious.

 

[MarcL]

not at all, my assumption of n being even was just wrong, but that's all I've been taught before so I thought it was a set definition. anyway case closed.

 

[phinds]

not at all, my assumption of n being even was just wrong, but that's all I've been taught before so I thought it was a set definition. anyway case closed.

You misunderstand. You copied the problem incorrectly. Do you not see the obvious mistake?

 

[MarcL]

I somehow read d... definitely sorry about that. I'll go crawl in a hole now.

 

[Merlin3189]

" I'll go crawl in a hole now." Can't see any reason to do that! Just change c) to n³ is odd , then carry on. No need to worry about a trivial typo.

So, " n=2k therefore n=(2k)² " what makes you say that?
It's just like saying n=n² which is not generally true.