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### What is a Property?

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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
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```
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
things you end up with (theorems).

more, or if you have any other questions.

- Doctor Ian, The Math Forum
http://mathforum.org/dr.math/
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Associated Topics:
High School Definitions
High School Logic

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