Indexed Union and Intersection

Families of relations and sets

Warning

Please pay attention to the type of the FF parameter in each definition.

The math notation (as is typical) hides the typing and leaves it for the reader to determine on their own. As such, it can be challenging to know whether you are indexing a family of sets or relations. Hopefully, this page will help clear that up.

Indexed Union

Family of Relations

Cupā€…ā€ŠIā€…ā€ŠAā€…ā€ŠBā€…ā€ŠF:={ā€…ā€Šx∈A,y∈B:∃(i∈I::(x,y)∈F[i])ā€…ā€Š}\textbf{Cup} \; I \; A \; B \; F := \{\; x \in A, y \in B : \exists(i \in I :: (x,y) \in F[i]) \;\}

I,A,B∈SetI, A, B \in \textbf{Set}

F∈(A→B)IF \in (A \to B)^I


Notation.

  1. Cupā€…ā€ŠIā€…ā€ŠAā€…ā€ŠBā€…ā€ŠF\textbf{Cup} \; I \; A \; B \; F can be written as Cupā€…ā€ŠIā€…ā€ŠF\textbf{Cup} \; I \; F when AA and BB are clear from the context.

  2. Cupā€…ā€ŠIā€…ā€ŠF\textbf{Cup} \; I \; F is written with symbols as ā‹ƒi∈IFi\bigcup_{i \in I} F_i

  3. ā‹ƒi∈IFi\bigcup_{i \in I} F_i can be written as ā‹ƒ(i:i∈I:F.i)\bigcup( i : i \in I: F.i)


fun Cup(I,A,B: set univ, F: I->A->B) : A->B {
  { x: A, y: B | some i: I | x->y in i.F }
}

Family of Sets

Cupā€…ā€ŠIā€…ā€ŠAā€…ā€ŠF:={ā€…ā€Šx∈A:∃(i∈I::x∈F[i])ā€…ā€Š}\textbf{Cup} \; I \; A \; F := \{\; x \in A : \exists(i \in I :: x \in F[i]) \;\}

I,A∈SetI, A \in \textbf{Set}

F∈I→AF \in I \to A


Notation.

  1. Cupā€…ā€ŠIā€…ā€ŠAā€…ā€ŠF\textbf{Cup} \; I \; A \; F can be written as Cupā€…ā€ŠIā€…ā€ŠF\textbf{Cup} \; I \; F when AA is clear from the context.

  2. Cupā€…ā€ŠIā€…ā€ŠF\textbf{Cup} \; I \; F is written with symbols as ā‹ƒi∈IFi\bigcup_{i \in I} F_i

  3. ā‹ƒi∈IFi\bigcup_{i \in I} F_i can be written as ā‹ƒ(i:i∈I:F.i)\bigcup( i : i \in I: F.i)


fun Cup(I,A: set univ, F: I->A) : set A {
  { x: A | some i: I | x in i.F }
}

Indexed Intersection

Family of Relations

Capā€…ā€ŠIā€…ā€ŠAā€…ā€ŠBā€…ā€ŠF:={ā€…ā€Šx∈A,y∈B:āˆ€(i∈I::(x,y)∈F[i])ā€…ā€Š}\textbf{Cap} \; I \; A \; B \; F := \{\; x \in A, y \in B : \forall(i \in I :: (x,y) \in F[i]) \;\}

I,A,B∈SetI, A, B \in \textbf{Set}

F∈(A→B)IF \in (A \to B)^I


Notation.

  1. Capā€…ā€ŠIā€…ā€ŠAā€…ā€ŠBā€…ā€ŠF\textbf{Cap} \; I \; A \; B \; F can be written as Capā€…ā€ŠIā€…ā€ŠF\textbf{Cap} \; I \; F when AA and BB are clear from the context.

  2. Capā€…ā€ŠIā€…ā€ŠF\textbf{Cap} \; I \; F is written with symbols as ā‹‚i∈IFi\bigcap_{i \in I} F_i

  3. ā‹‚i∈IFi\bigcap_{i \in I} F_i can be written as ā‹‚(i:i∈I:F.i)\bigcap( i : i \in I: F.i)


fun Cap(I,A,B: set univ, F: I->A->B) : A->B {
  {x: A, y: B | all i: I | x->y in i.F }
}

Family of Sets

Capā€…ā€ŠIā€…ā€ŠAā€…ā€ŠF:={ā€…ā€Šx∈A:āˆ€(i∈I::x∈F[i])ā€…ā€Š}\textbf{Cap} \; I \; A \; F := \{\; x \in A : \forall(i \in I :: x \in F[i]) \;\}

I,A∈SetI, A \in \textbf{Set}

F∈I→AF \in I \to A


Notation.

  1. Capā€…ā€ŠIā€…ā€ŠAā€…ā€ŠF\textbf{Cap} \; I \; A \; F can be written as Capā€…ā€ŠIā€…ā€ŠF\textbf{Cap} \; I \; F when AA is clear from the context.

  2. Capā€…ā€ŠIā€…ā€ŠF\textbf{Cap} \; I \; F is written with symbols as ā‹‚i∈IFi\bigcap_{i \in I} F_i

  3. ā‹‚i∈IFi\bigcap_{i \in I} F_i can be written as ā‹‚(i:i∈I:F.i)\bigcap( i : i \in I: F.i)


fun Cap(I,A: set univ, F: I->A) : set A {
  { x: A | all i: I | x in i.F }
}

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