Prove/Disprove: if $x^2 = a^2$, then $x = a$ From Prof. Charles Pinter's "A Book of Abstract Algebra"'s Chapter 4 exercises:


  For each of the following rules, either prove that it is true in every group $G$, or give a counter-example.


$$ \text{if } x^{2}=a^{2}, \text{then } x=a$$

I believe that this is true by:

$$xx = aa$$

by cancellation,

$$x=a$$

Is that right? Also, when the word "prove" is used, does that mean to use theorems to prove?
What is it that you are canceling there? It looks to me like you used that $x=a$ to show that $x=a$, which is not a good plan.

Think: is this true even in the real numbers?
Cancellation is not possible, you must take the root of both sides. Square roots have the property $q = \pm\sqrt{q^2}$, therefore $a = \pm x$, which is not equivalent to $a = x$. The original statement has thus been disproved.