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]AB=\{x\in A\vert x\notin B\}[/ilmath]
Alternative forms
 [ilmath]AB=(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]XA[/ilmath], so if we have [ilmath]A,B\in\mathcal{P}(X)[/ilmath] then [ilmath]AB[/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](AB)C=A(B\cup C)[/ilmath]
Proof:
 Note that [ilmath]AB=(A^c\cup B)^c[/ilmath] so [ilmath](AB)C = ((AB)^c\cup B)^c =(((A^c\cup B)^c)^c\cup C)^c[/ilmath]
 But: [ilmath](A^c)^c=A[/ilmath] so:
 [ilmath](AB)C=(A^c\cup B\cup C)^c=(A^c\cup(B\cup C))^c=A(B\cup C)[/ilmath]
 But: [ilmath](A^c)^c=A[/ilmath] so:
TODO: Make this proof neat
See also
References
 ↑ ^{1.0} ^{1.1} Measure Theory  Paul R. Halmos

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