I am currently reading:
How to think like a mathematician By Kevin Houston
Early in the book it defines difference ( in sets) in terms of $X$ and $Y$ and then states that is if $Y$ is a subset of $X$ we can call $X\setminus Y$ the complement of $Y$ in $X$ and denote this by $Y^{c}$
Later on in the book it uses this complement notation more and I was just curious to know how you could refer to which set the complement was in if say $Y$ was a subset of $A$ and $B$ how would write these individually?
When $Y$ is a subset of $A$ the complement is $A\backslash Y$. You use the complement notation $Y^c$ only when you want the complement in the universe of discourse at the moment: the set of everything you are talking about at the moment. So in the integers, $\text{odds}^c = \text{evens}$ but in the real numbers you'd have to write $\text{integers}\backslash \text{odds}$ since $\text{odds}^c$ would be all the real numbers that weren't odd integers.