I'm working on addition and multiplication axioms of integers for discrete math. I'm trying to prove (k - m) + (m - n) = k - n. The first step I took was this (without citing any of the axioms):
= k + (-m) + m + (n)
Is this incorrect?
On a related note, the solution in the textbook on the other hand starts off like this:
= (k + (-1)m) + (m + (-1)n)
= k + [(-1)m + m + (-1)n)
After that, (-1)m + m is zero. What I don't understand is how do they go from step 1 to step 2?
The axioms I'm given are as follows:
We start with $(k-m)+(m-n)$, now we start with the first step you mentioned:
1) $\big(k+(-1)m\big)+\big(m+(-1)n\big)$
For clarity, let's denote $m+(-1)n=a$, then we have $\big(k+(-1)m\big)+a$.
2) $\big(k+(-1)m\big)+a = k+\big((-1)m+a\big)$ by associativity.
Finally, we have $$k+\big((-1)m+a\big) = k+\big((-1)m+m+(-1)n\big) = k+\big(0+(-1)n\big) = k-n. $$