Set subtraction

(Unknown grade)

This page is a stub

This page is a stub, so it contains little or minimal information and is on a to-do list for being expanded.The message provided is:

Cleanup and further expansion

(Unknown grade)

This page requires references, it is on a to-do list for being expanded with them.

Please note that this does not mean the content is unreliable, it just means that the author of the page doesn't have a book to hand, or remember the book to find it, which would have been a suitable reference.

Definition

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

Be bothered to do this

Terminology

• Relative complement
  • 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

---

See also

• Set complement