Prove that $(p\oplus q)\oplus r$ is logically equivalent to $p\oplus (q \oplus r)$ I am reviewing for my upcoming Discrete Mathematics class and I'm clueless on the aforementioned question. I've tried expounding the XOR operator in hopes of transforming the problem but still I cannot find the solution to the problem.
The title is pretty self-explanatory, how do I prove their logical equivalence without using a truth table?
The truth table of the problem looks like this:
Problem truth table


XOR evaluates to true when there are "odd" number of true inputs. So the order and association don't matter. You may think of XOR as$\mod 2$ operator. As an example,
$$odd + even +even =  odd $$ 
It doesn't matter how you order or associate the inputs.
