Well I've got a pretty lengthy study guide for an exam next week and I have a few questions that I get a little stuck on... Any help, with our without an explanation (with is always best!) would be appreciated.

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The converse and contrapositive of this:

x^2 = 1 → x = ±1

I'm not sure if I make everything a ¬ or just what is between the = signs. Do I make the 1 a 0? Does the 1 imply a tautology or is it simply a constant 1?

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I need to write this statement using the quantifiers "for all (or for every)" and "there exists".

An integer cannot be both even and odd.

What I could muster up was "For all integers, there exists only an odd or only an even". This really doesn't make much sense, but I really couldn't get much out of it.

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Negations. I never know what I need to negate...

Every integer is divisible by a prime. Would this be "Not every integer is divisible by a prime"?

If n is an integer, n/(n+1) is not an integer. Would this be "If n is not an integer, n/(n+1) is not an integer"? I could use an explanation for this one!

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And finally....

I need to answer true or false and supply a direct proof or a counter-example to each of the following:

There exists an integer n ≠ 0 such that nq is an integer for every rational number q.

and

For every rational number q, there exists an integer n ≠ 0 such that nq is an integer.

To be honest, I can't even begin to answer these. I'm totally lost.

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I know this is a lot to throw at you guys, but any help is GREATLY appreciated. I'm so horrible at math, and my teacher honestly doesn't know discrete math. He's teaching this for the first time because the school couldn't find any other teachers.

Thanks everyone in advance!