Difference between revisions of "Set subtraction"
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Latest revision as of 00:48, 21 March 2016
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Contents
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]
- But: [ilmath](A^c)^c=A[/ilmath] so:
TODO: Make this proof neat
See also
References
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Categories:
- Stub pages
- Pages requiring references
- Pages requiring references of unknown grade
- Pages requiring proofs
- Pages requiring proofs of unknown grade
- Todo
- Definitions
- Set Theory Definitions
- Set Theory
- Theorems
- Theorems, lemmas and corollaries
- Set Theory Theorems
- Set Theory Theorems, lemmas and corollaries
- Set operations
