Set subtraction

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Definition

Given two sets, $A$ and $B$ we define set subtraction (AKA: relative complement) as follows:

• $A-B=\{x\in A\vert x\notin B\}$

Alternative forms

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• $A-B=(A^c\cup B)^c$

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Terminology

• Relative complement
  • This comes from the idea of a complement of a subset of $X$, say $A$ being just $X-A$, so if we have $A,B\in\mathcal{P}(X)$ then $A-B$ can be thought of as the complement of $B$ if you consider it relative (to be in) $A$.

Notations

Other notations include:

• $A\setminus B$

Trivial expressions for set subtraction

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Claim: $(A-B)-C=A-(B\cup C)$

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Proof:

• Note that $A-B=(A^c\cup B)^c$ so $(A-B)-C = ((A-B)^c\cup B)^c =(((A^c\cup B)^c)^c\cup C)^c$
  • But: $(A^c)^c=A$ so:
    • $(A-B)-C=(A^c\cup B\cup C)^c=(A^c\cup(B\cup C))^c=A-(B\cup C)$

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TODO: Make this proof neat

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See also

• Set complement