The Strongest Matthew For Whenever Options

[Dr Strum]

Back for it's third iteration, and still longer than any other.
You know what to do. Just post about stuff. Your life, your insanity, or even your favorite cardinal direction.

[Elnendil]

Definitions can't have proofs. Proofs are for statements, and are based off assumptions. Definitions are just notation, a shorthand for a property or set of properties. It's like proving that 1+1 = 2. The definition of 2 is 1 + 1. You can't prove anything about 2 until you already know what that symbol means, and what it means is 1+1. So you can't prove 1+1 = 2.

What you can do is prove that different definitions are EQUIVALENT. If you have property X with the name N (i.e. N is defined as "has property X"), and you prove that property X holds if and only if property Y holds, then Y is also a valid definition for N. But again, you have to have a definition to prove that other things are equivalent to it.

You'll have to be more precise with your language. I still don't know what you're trying to say.

I'll just make this point I guess. The book that i'm using in question doesn't call what we've been discussing definitions, and goes out of its way to prove the statements (If a|b and a|c, then a|(bx+cy) for some x and y is defined as a theorem in my book and has a proof for it). There are actual "definitions" in the book, like d=(a,b), and thats stupid to prove because its a definition of what the greatest common divisor is, its just the notation you're supposed to use. Thats really about in on me using the definition here. If you're saying that the concept is so simple that its practically the definition, I suppose I can agree there.

Aside from that, like I said before, i'm going to stick to the book.

I'm NOT dismissing anything. I'm trying to figure out WHAT YOU'RE TRYING TO SAY, because what you're saying doesn't make sense to me in the literal sense, but I'm not sure if it's because you don't know the notation that I'm used to. I'm trying to eliminate possibilities.

The reason behind me saying i'll just stick to the book and asking questions is because i'm assuming you've had more mathematical experience than I have (I'm assuming you've passed Number Theory). This is also one of my first proof courses, so proving a lot of statements I haven't considered to prove is somewhat new to me. Thats probably why i'm not making much sense to you, alongside possibly differences in how our professors and the book tackled the concept for us.

This is the book i've been using: http://www.amazon.com/Elementary-Introduct...n/dp/0881338362