I'm familiar with XOR in logical terms, meaning $A \oplus B$ meaning mutual exclusiveness, and I can easily write out the truth table for it.
But what does the XOR operator mean when it's used on numbers in the decimal system, and not binary?
From examples, I've seen that
$10 \oplus 10 = 0$,
$10 \oplus 11 = 1$,
$10 \oplus 12 = 6$.
This, I don't understand, and from further examples, I'm not seeing a pattern.
Could anyone explain how one would reach these results?
This is just the bitwise XOR. That means write each number out in binary, and XOR the corresponding bits.
For $10 \oplus 12$, note that $10 = 1010_2$ and $12 = 1100_2$. So, we have the following:
1st bit: $1 \oplus 1 = 0$
2nd bit: $0 \oplus 1 = 1$
3rd bit: $1 \oplus 0 = 1$
4th bit: $0 \oplus 0 = 0$
Putting this together, we get $1010_2 \oplus 1100_2 = 0110_2$, i.e. $10 \oplus 12 = 6$. I'll let you work out the other examples.