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I'm a first-year student and I am in advanced calculus (specifically, Analysis I). I, however, switched into the class during the third week. So I am having quite a bit of difficulty, because I haven't learned any fundamental concepts.

So I am asking if someone can please discuss a couple of concepts/questions with me:

-How to prove uniqueness (specifically with inequalities)

-I know about induction, but I'm not sure how to use it to prove inequalities

-inf/sup/max/min

-identity function and right/left inverse

And for this question, I do not understand it at all:

"1a) let k be any natural number. use induction to prove: theorem

for any integer n, which is an integer, there is a unique q, which is an integer, such that 0 <= n+qk <k

(the resulting number n + qk {0,.....,k-1} is somtines denoted "n mod k")

b)

for any n that is an integer (I have to be explicit, because I don't know how to type those math symbols) let [n] C Integers, be the subset.

[n] = {n+qk, q is any integer}

note that

[n + pk] = [n] for any n,p that are integers. hence, by part a) there are actually just k distinct subsets, [0],...,[k-1]. Let Z(subscript k) be the set of such subsets,

z(subscript k) = {[0],...,[k-1]}.

define an addition and a multiplication on z(subscript k) by

[n] + [m] = [n+m], [n][m]=[nm]

show that z(subscript 3) satifies the axioms (p1) to (p9), but z (subscript 4) does not. (don't have to write out the proof that these formulas for + and * are well-defined)."

the axioms are from my spivak textbook. can someone just discuss what is going on in this question.

Thanks for any help.

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# Conceptual questions.

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