# Set subtraction

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## Definition

Given two sets, [ilmath]A[/ilmath] and [ilmath]B[/ilmath] we define set subtraction (AKA: relative complement[1]) as follows:

• [ilmath]A-B=\{x\in A\vert x\notin B\}[/ilmath]

### Alternative forms

• [ilmath]A-B=(A^c\cup B)^c[/ilmath]

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## Terminology

• Relative complement[1]
• This comes from the idea of a complement of a subset of [ilmath]X[/ilmath], say [ilmath]A[/ilmath] being just [ilmath]X-A[/ilmath], so if we have [ilmath]A,B\in\mathcal{P}(X)[/ilmath] then [ilmath]A-B[/ilmath] can be thought of as the complement of [ilmath]B[/ilmath] if you consider it relative (to be in) [ilmath]A[/ilmath].

## Notations

Other notations include:

• [ilmath]A\setminus B[/ilmath]

## Trivial expressions for set subtraction

Claim: [ilmath](A-B)-C=A-(B\cup C)[/ilmath]

Proof:

• Note that [ilmath]A-B=(A^c\cup B)^c[/ilmath] so [ilmath](A-B)-C = ((A-B)^c\cup B)^c =(((A^c\cup B)^c)^c\cup C)^c[/ilmath]
• But: [ilmath](A^c)^c=A[/ilmath] so:
• [ilmath](A-B)-C=(A^c\cup B\cup C)^c=(A^c\cup(B\cup C))^c=A-(B\cup C)[/ilmath]

TODO: Make this proof neat