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It is no coincidence that the set difference $A\setminus B =\{x \mid x\in A \wedge x\notin B\}$ and the set complement $B^C =\{x \mid x\in A\wedge x\notin B\}$ have the same set-builder notations. In fact, they are almost the same mathematical concepts. Certainly, they are equal sets $B^C=A\setminus B$, and we therefore also call the set difference $A\setminus B$ the “relative complement of $B$ with respect to $A$”. The reason why mathematicians use a separate name “complement” and notation $B^C$ instead of just talking about the set difference $A\setminus B$ is that sometimes, the set $A$ is so clear from the context that makes a line of thought clearer to leave $A$ out. This is another important consequence of the axiom of separation:

Corollary: Set Difference and Set Complement are the Same Concepts

In the Zermelo-Frankel set theory, no set $B$ has an “absolute complement” $B^C$. This set complement is equal to the set difference $A\setminus B$ being the relative complement of $B$ with respect to $A$. This is true even if we take $A$ as the universal set.

| | | | | Contributors: bookofproofs | References: [656], [983]

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