Algebraic Systems

Date: 09/07/97 at 16:31:04
From: matt hoffman
Subject: Algebraic systems

A binary operation * is defined for real numbers. For the equation 
a*b = b how do you determine if * is associative or commutative?
The same for this equation: a*b = ab+1 .

I thought I understood that a+b = b+a and a*b = b*a for the 
commutative property and that for the associative property 
(a+b)+c = a+(b+c) but I don't know how to explain for the others.

Thank you.

Date: 09/13/97 at 23:42:44
From: Doctor Mike
Subject: Re: Algebraic systems

Hi Matt,    

You understood correctly.... sort of.  

If "+" means normal addition and "*" means normal multiplication, 
then both of these well-known operations are commutative and 
associative.

The idea of this exercise is for you to test some new made-up,
never-before-heard-of operations, to see if they have the same
properties as the standard operations.  That will help you to 
understand these properties in their pure abstract form.
   
That's why it is best for understanding if you use a completely
different symbol, like "&" for instance.  Like this : 
   
A binary operation & is defined for real numbers.  If the 
definition of this operation is a&b = b, tell whether & is 
commutative.
   
It is not, as a simple example shows: 4&7 = 7  but 7&4 = 4
   
What about associativity for &?  Let's see what that would mean.
    
     Is it true that  (a&b)&c = a&(b&c)   ?
     That's the same as (b)&c = a&(c)
     That's the same as     c = c
   
So, this made-up operation & is associative, but not commutative.
   
Now you try it for the operation "$" defined by a$b = a*b+1 .  
I hope this helps you get started with this stuff.  Good luck.

-Doctor Mike,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/