Complement

WHAT IS LEFT WHEN A SUBSET IS REMOVED

Removing a subset from the set it sits in leaves the rest behind, and the rest is a subset too. That leftover is the complement. It is not a property of the subset on its own: the same collection of points has different leftovers depending on what it is being removed from, so the complement is always taken with respect to a stated set.

one subset V = {1}, ringed, sitting inside two different M M = {1,2} 1 2 V M ∖ V = {2} M = {1,2,3} 1 2 3 V M ∖ V = {2,3} V never changed; its complement did
Brown is \(V\), blue is \(M \setminus V\). On each row the two together are exactly \(M\).

The definition

Definition · complement

Let \(V \subseteq M\). The complement of \(V\) in \(M\) is:

\[ M \setminus V := \{\, p \in M \mid p \notin V \,\}. \]

Taking the complement twice returns what you started with, \(M \setminus (M \setminus V) = V\), so the operation is its own inverse. And it exchanges the two extremes: \(M \setminus \emptyset = M\) and \(M \setminus M = \emptyset\).

The ambient set is part of the question

Nothing about \(V\) differs between the two rows above, and the two answers do. So "the complement of \(V\)" is an incomplete phrase until \(M\) is named.

The notation carries both for that reason: \(M \setminus V\) names the set the points are drawn from as well as the part being removed.