Set subtraction
From Maths
Contents
Definition
Given two sets, [ilmath]A[/ilmath] and [ilmath]B[/ilmath] we define set subtraction as follows:
- [ilmath]A-B=\{x\in A\vert x\notin B\}[/ilmath]
Equivalent definitions
- [ilmath]A-B=(A^c\cup B)^c[/ilmath]
TODO: Be bothered to do this
Other names
- Relative complement
- This comes from the fact that the complement of a subset of [ilmath]X[/ilmath], [ilmath]A[/ilmath] is just [ilmath]X-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
TODO: Find references
