Discrete Math Proof: Proving Equivalence of 4 Statements

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SUMMARY

The forum discussion centers on proving the equivalence of four statements regarding the parity of integers: (a) n² is odd, (b) 1 - n is even, (c) n² is odd, and (d) n² + 1 is even. A participant initially misinterpreted the problem due to a typographical error, confusing the statements. The resolution involved correcting statement (c) to n³ is odd, allowing for a valid proof of equivalence among the statements. The discussion highlights the importance of accurate problem transcription in mathematical proofs.

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MarcL
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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?
 
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MarcL said:
(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?
 
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
 
MarcL said:
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.
 
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.
 
MarcL said:
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?
 
I somehow read d... definitely sorry about that. I'll go crawl in a hole now.
 
" 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.
 

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