Relative complement

The relative complement of two sets \(A\) and \(B\), denoted \(A \setminus B\), is the set of elements that are in \(A\) while not in \(B\).

illustration of the output of a relative complement

Formally stated, where \(C = A \setminus B\)

$$x \in C \leftrightarrow (x \in A \land x \notin B)$$

That is, Iff \(x\) is in the relative complement \(C\), then \(x\) is in \(A\) and x is not in \(B\).

For example,

  • \(\{1,2,3\} \setminus \{2\} = \{1,3\}\)

  • \(\{1,2,3\} \setminus \{9\} = \{1,2,3\}\)

  • \(\{1,2\} \setminus \{1,2,3,4\} = \{\}\)

If we name the set \(U\) as the set of all things, then we can define the Absolute complement of the set \(A\), \(A^\complement\), as \(U \setminus A\)