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### 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
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/
```
Associated Topics:
High School Basic Algebra

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