Abstract Algebra IBL Wiki

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====== Why we need associative ======

FIXME Add in the axioms of multiplication, as well as a few basic rules about solving equations (such as if $x=y$, then $ax=ay$.  

==== Problem ====

Consider the equation $2x=3$.  Carefully solve this problem for $x$, where at each stage of your computation you should explain what rule you are using to performing.  Do not skip any steps in your computation. 

Did you use the commutative rule of multiplication above? Look through your work above for the commutative rule. If you used it, please rewrite your work so that it does not use the commutative rule (it's not needed).

Now go back through your work above and look for where you used the associative rule of multiplication. If you didn't use it, then check your work again, as you have definitely skipped explaining some step in your computation.


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==== Remarks ====  

  *This problem shows a student how crucial associativity is. 
  *It might be best to point them to a list of axioms, or something they can draw upon for their explanations. 
  *I was actually hoping to just have them tackle this one with small group presentations, then have a lively discussion about how detailed proofs should be, and then wrap it up as a class (pointing out how engrained associativity is in everything we do).  
  *As the semester unfolds, we can come back to the associative laws in various other settings.  One question will be, "What does associativity mean in a Cayley Graph?"

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==== $\LaTeX$ version ====
<code>
%%%%%%%%%%
% DEPENDENCIES
% RequiredPackages: \usepackage{tikz}
% RequiredMacros: \DeclareMathOperator{\aut}{Aut} 
%%%%%%%%%%
\begin{problem}
Type the problem code here.
\end{problem}
</code>

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==== External links ====
  *http://www.math.sjsu.edu/~hsu/courses/131a/axioms.pdf A list of axioms
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