Hi!

And here is a little different way to look at commutativity and associativity.

Commutativity can be viewed more as a language problem than an axiom problem.  Let me
explain.  Take the example of adding four and five.  We typically write and speak in a linear array
of letters or syllables.  This forces us to say one of the numbers first and the other second.  So
we say "four plus five" or "five plus four" and write 4+5 or 5+4.  But we want these two
expressions to mean the same thing; that is, we want them to get us to "9."

As a function (binary operator in this case) we have a function "+" which maps (4,5) to 9 and
also (5,4) to 9.  If we had a way to "say" the four and five SIMULTANEOUSLY, then we would
have but one way to "speak" or "write" about the addition of four and five.  Thus we wouldn't
have to "invoke" a commutativity rule.

Actually we could define addition of 4 and 5 as a mapping from the SET {4,5} to 9.  Since the
set is unordered this gives us only one way to express the combination of the 4 and 5 to get
the 9.  Then "4+5" and "5+4" could be viewed as ways of saying that {4,5} maps to 9 or that
these are ways of expressing the result of mapping {4,5} to 9.  But then the notation +{4,5}=9
is a bit awkward compared to 4+5=9 or 5+4=9.  On the other hand as the sets get larger, for
example {3,4,5}, +{3,4,5}=12 is not so bad.  And here with the three numbers using this
approach we don't have to get into associativity either.

If we define 3, 4, and 5 by 3={ooo}, 4={oooo} and 5={ooooo} (multisets) then adding these
three sets together is just in essence dumping them altogether to get {oooooooooooo}.  How
we dump them together (all at once, one at a time, etc.) is immaterial.  We get the same result.

So commutativity and associativity can be view more as a language problem than an axiom
problem.  On the other hand, the additive and multiplicative identity and inverse axioms are
axiom problems in the sense that they do not just involve ways of "saying the same thing".
They inject the notion of "existence" into the system.

All this being said, viewing the way our systems of numbers have evolved, it is perhaps easier
to just introduce commutativity and associativity as axioms and be done with it.

Have a very blessed day!

anonimnystefy wrote:

It cannot be prpved. It is an axiom of natural numbers and of higher number sets, too.

It depends on how you are defining your sets of numbers. If you are defining them by axioms, then there is nothing to prove since the axioms will include commutativity and associativity of addition. But if you are constructing them from smaller sets of numbers, then commutativity and associativity need to be proved.

For example, here is briefly how

is constructed from

. Define a relation

on

by

iff

. (Hint: Think of

as the "difference"

.) Then

is an equivalence relation and

is defined as the set of all equivalence classes under

. Let the equivalence class containing

be denoted

. Then addition in

is defined as

After checking that the operation is well defined, one can proceed to verify that

is commutative and associative.

Moreover, zero

in

is the equivalence class

and multiplication in

is defined as

.

Similarly

can be constructed from

as equivalence classes of the equivalence relation

on

defined by

iff

. Letting the equivalence class containing

be denoted

addition in

is definied as

Last edited by scientia (2013-02-18 23:19:54)

No,

is constructed by means of either Dedekind cuts or equivalence classes of Cauchy sequences of rationals. The process is very different and much more complicated because you are now constructing an uncountable set from a countable one.