Difference between revisions of "Set subtraction"
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==Definition== | ==Definition== | ||
| − | Given two sets, {{M|A}} and {{M|B}} we define ''set subtraction'' as follows: | + | Given two sets, {{M|A}} and {{M|B}} we define ''set subtraction'' ({{AKA}}: ''relative complement''{{rMTH}}) as follows: |
* {{M|1=A-B=\{x\in A\vert x\notin B\} }} | * {{M|1=A-B=\{x\in A\vert x\notin B\} }} | ||
| − | == | + | ===Alternative forms=== |
| − | * Relative complement | + | {{Begin Inline Theorem}} |
| − | ** This comes from the | + | * {{M|1=A-B=(A^c\cup B)^c}} |
| + | {{Begin Inline Proof}} | ||
| + | {{Requires proof|Be bothered to do this}} | ||
| + | {{End Proof}}{{End Theorem}} | ||
| + | ==Terminology== | ||
| + | * '''Relative complement'''<ref name="MTH"/> | ||
| + | ** This comes from the idea of a [[complement]] of a subset of {{M|X}}, say {{M|A}} being just {{M|X-A}}, so if we have {{M|A,B\in\mathcal{P}(X)}} then {{M|A-B}} can be thought of as the complement of {{M|B}} if you consider it relative (to be in) {{M|A}}. | ||
==Notations== | ==Notations== | ||
Other notations include: | Other notations include: | ||
* {{M|A\setminus B}} | * {{M|A\setminus B}} | ||
| − | == | + | ==Trivial expressions for set subtraction== |
{{Begin Inline Theorem}} | {{Begin Inline Theorem}} | ||
| − | + | '''Claim:''' {{M|1=(A-B)-C=A-(B\cup C)}} | |
{{Begin Inline Proof}} | {{Begin Inline Proof}} | ||
| − | {{Todo| | + | '''Proof:''' |
| + | * Note that {{M|1=A-B=(A^c\cup B)^c}} so {{M|1=(A-B)-C = ((A-B)^c\cup B)^c =(((A^c\cup B)^c)^c\cup C)^c}} | ||
| + | ** But: {{M|1=(A^c)^c=A}} so: | ||
| + | *** {{M|1=(A-B)-C=(A^c\cup B\cup C)^c=(A^c\cup(B\cup C))^c=A-(B\cup C)}} | ||
| + | {{Todo|Make this proof neat}} | ||
{{End Proof}}{{End Theorem}} | {{End Proof}}{{End Theorem}} | ||
| + | |||
==See also== | ==See also== | ||
* [[Set complement]] | * [[Set complement]] | ||
==References== | ==References== | ||
<references/> | <references/> | ||
| − | {{ | + | {{Set operations navbox|plain}} |
{{Definition|Set Theory}} | {{Definition|Set Theory}} | ||
{{Theorem Of|Set Theory}} | {{Theorem Of|Set Theory}} | ||
| + | [[Category:Set operations]] | ||
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]
(Unknown grade)
This page requires one or more proofs to be filled in, it is on a to-do list for being expanded with them.
Please note that this does not mean the content is unreliable. Unless there are any caveats mentioned below the statement comes from a reliable source. As always, Warnings and limitations will be clearly shown and possibly highlighted if very important (see template:Caution et al).
The message provided is:
The message provided is:
Be bothered to do this
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
