What is a Property?
Date: 11/29/2001 at 11:14:54 From: Jim Lyons Subject: Properties, what exactly are they? I understand what Undefined Terms and Defined terms are. I also understand what Axioms and Theorems are, but what exactly is a Property? Is it the same thing as a Theorem? Also, what is a Law? The Distribution Property is sometimes referred to as the Distribution Law in Math books. Thanks, Jim
Date: 11/29/2001 at 12:10:35 From: Doctor Ian Subject: Re: Properties, what exactly are they? Hi Jim, An axiom is a statement that is assumed to be true in some formal system. A theorem is a statement that is provable, given a set of axioms and proof procedures. An axiom or theorem _in_ a given system is often said to be a property _of_ that system. For example, in standard number theory, the axiom (a + b) + c = a + (b + c) for any numbers a and b might be either an axiom or a theorem (depending on which formulation you're using); we call it 'associativity of addition'; and we say that 'associativity of addition' is a property of numbers. In math, 'law' is yet another way to say 'property', but implies that you're looking at it in a slightly different way. It's more natural to think of a property as an 'axiom' or 'theorem' when you're using it to prove more theorems. It's more natural to think of it as a 'property' when you're trying to use it to solve a particular problem (that is, you're trying to figure out what your next step should be). It's more natural to think of it as a 'law' when you're trying to check the solution to a problem (that is, you don't want to accept a solution unless all the relevant 'laws' have been 'obeyed'). But in the end, they're all just axioms and theorems, and the only real distinction is between things that you start with (axioms) and things you end up with (theorems). I hope this helps. Write back if you'd like to talk about this some more, or if you have any other questions. - Doctor Ian, The Math Forum http://mathforum.org/dr.math/
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