Indexed Union and Intersection

Families of relations and sets

Warning

Please pay attention to the type of the $F$ 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

$\textbf{Cup} \; I \; A \; B \; F := \{\; x \in A, y \in B : \exists(i \in I :: (x,y) \in F[i]) \;\}$

---

$I, A, B \in \textbf{Set}$

$F \in (A \to B)^I$

---

Notation.

1. $\textbf{Cup} \; I \; A \; B \; F$ can be written as $\textbf{Cup} \; I \; F$ when $A$ and $B$ are clear from the context.

2. $\textbf{Cup} \; I \; F$ is written with symbols as $\bigcup_{i \in I} F_i$

3. $\bigcup_{i \in I} F_i$ can be written as $\bigcup( i : i \in I: F.i)$

---

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

$\textbf{Cup} \; I \; A \; F := \{\; x \in A : \exists(i \in I :: x \in F[i]) \;\}$

---

$I, A \in \textbf{Set}$

$F \in I \to A$

---

Notation.

1. $\textbf{Cup} \; I \; A \; F$ can be written as $\textbf{Cup} \; I \; F$ when $A$ is clear from the context.

2. $\textbf{Cup} \; I \; F$ is written with symbols as $\bigcup_{i \in I} F_i$

3. $\bigcup_{i \in I} F_i$ can be written as $\bigcup( i : i \in I: F.i)$

---

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

$\textbf{Cap} \; I \; A \; B \; F := \{\; x \in A, y \in B : \forall(i \in I :: (x,y) \in F[i]) \;\}$

---

$I, A, B \in \textbf{Set}$

$F \in (A \to B)^I$

---

Notation.

1. $\textbf{Cap} \; I \; A \; B \; F$ can be written as $\textbf{Cap} \; I \; F$ when $A$ and $B$ are clear from the context.
2. $\textbf{Cap} \; I \; F$ is written with symbols as $\bigcap_{i \in I} F_i$
3. $\bigcap_{i \in I} F_i$ can be written as $\bigcap( i : i \in I: F.i)$

---

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

$\textbf{Cap} \; I \; A \; F := \{\; x \in A : \forall(i \in I :: x \in F[i]) \;\}$

---

$I, A \in \textbf{Set}$

$F \in I \to A$

---

Notation.

1. $\textbf{Cap} \; I \; A \; F$ can be written as $\textbf{Cap} \; I \; F$ when $A$ is clear from the context.
2. $\textbf{Cap} \; I \; F$ is written with symbols as $\bigcap_{i \in I} F_i$
3. $\bigcap_{i \in I} F_i$ can be written as $\bigcap( i : i \in I: F.i)$

---

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