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These are the course notes for MIT's calculus class, 18.014.

Beginning at the bottom of page 4, the professor launches into a long proof that the set of all integers $\mathbb Z$ is closed under addition.

I don't see why I can't demonstrate it by reasoning the following way. All integers in $\mathbb Z$ are either elements of $P$ (the set of positive integers), the negative of $P$'s elements, or $0$. $P$ is defined as the set of elements common to all inductive sets. So $P$ therefore contains $1$ and $x+1$, for any $x$. Any numbers $a,b$ in $P$ is an integer by definition, and is composed of adding $1+1$ multiple times. Therefore, adding or subtracting $a$ and $b$ is really adding or subtracting $1$ multiple times.

Which means that the set of all integers is closed under addition and subtraction, since you can only get integers.

So why is the level of rigor in the lecture notes necessary?

  • 3
    You could enter that in the World's Longest Run-on Sentence competition. Or, you could edit it into something comprehensible.2012-08-19
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    It's readable now!2012-08-19
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    @GerryMyerson. First version of question was very clear. Sarcastical alusion to competition is impolite, specially for new user. Why you think you may speak like that to people here?2012-08-19
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    In fact, I don't understand why do you need to spend this amount of time and effort to prove such a thing. There are more interesting things at higher levels to work on them.2012-08-19
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    @evg, I figure anyone who takes the handle "terriblestudent" probably has a good enough sense of humor to take my comment in the light-hearted way it was intended.2012-08-19

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