I recently took a mathematical proof class and am beginning to teach myself abstract algebra. I'm fairly new to proofing however, and am not very confident in how I do it. Also, I'm new to this site so I apologize for any technical mistakes here (I ended up copying and pasting from Word…how do you do mathematical symbols and all on this site…I’m very low tech)
The book I am using is "A Book of Abstract Algebra" by Charles Pinter. This question is on pg 24 of Chapter 2(Operations) under exercise set D, question 1 for anyone who has the book and would like to reference it.
The question asks us to prove that the operation of concatenation is associative on $A^*$, the set of all sequences of symbols in the alphabet $A$. The operation, $*$ is defined as
$$ a * b := a_1 a_2…a_n b_1 b_2…b_m. $$
Theorem: The operation of concatenation denoted by $*$ and defined above is associative on $A^*$, the set of all sequences of symbols in the alphabet $A$.
My Proof: Suppose $x, y, z \in A$. Then $x*y=x_1 x_2…x_n y_1 y_2…y_m$.
Now call $x*y$, $a$.
Then $a*z=x_1 x_2…x_n y_1 y_2…y_m z_1 z_2…z_q$.
Since $a= x*y$, $a*z=(x*y)*z=x_1 x_2…x_n y_1 y_2…y_m z_1 z_2…z_q$.
Now $y*z=y_1 y_2…y_m z_1 z_2…z_q$.
Now call $y*z$, $b$.
Then $x*b=x_1 x_2…x_n y_1 y_2…y_m z_1 z_2…z_q$.
Since $b=y*z$, $x*b=x*(y*z)=x_1 x_2…x_n y_1 y_2…y_m z_1 z_2…z_q.=(xx_1 x_2…x_n y_1 y_2…y_m z_1 z_2…z_q=(x*y)*z$
Thus $\forall x \forall y \forall z \in A^*$, $(x*y)*z=x*(y*z)$, and hence by the definition of associativity, concatenation is associative on $A^*$.
So my questions are:
1) Is this proof correct (I mean this in a very basic sense, not is it written well, but is it correct; will it do?) If not, what do I need to do to fix it?
2) If it is correct, is there anything you'd do in terms of how it is written to make it better?