> For the complete documentation index, see [llms.txt](https://adams-personal-organization.gitbook.io/compendium-of-predicates/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://adams-personal-organization.gitbook.io/compendium-of-predicates/definitions/relation-inclusion.md).

# Relation Inclusion

<details>

<summary><span class="math">\textbf{Includes} \; A \; B \; R \; S</span></summary>

***

$$\textbf{Relation} ; A ; B ; R$$

$$\textbf{Relation} ; A ; B ; S$$

$$\forall (x \in A : \forall (y \in B : R.x.y : S.x.y))$$

***

**Notation.**

1. $$\textbf{Includes} ; A ; B ; R ; S$$ can be abbreviated by $$\textbf{Includes} ; R ; S$$when $$A$$ and $$B$$ are clear from the context.
2. $$\textbf{Includes} ; R ; S$$ can be written $$R \subseteq S$$.

***

```
pred Includes(A,B: set univ, R,S: univ->univ) {
  Relation[A,B,R]
  Relation[A,B,S]
  R in S
}
```

</details>

<details>

<summary><span class="math">\textbf{Within} \; A \; B \; R \; S</span></summary>

***

$$\textbf{Includes} ; A ; B ; S ; R$$

***

**Notation.**

1. $$\textbf{Within} ; A ; B ; R ; S$$ can be abbreviated by $$\textbf{Within} ; R ; S$$when $$A$$ and $$B$$ are clear from the context.
2. $$\textbf{Within} ; R ; S$$ can be written $$R \supseteq S$$.

***

```
pred Within(A,B: set univ, R,S: univ->univ) {
  Includes[A,B,S,R]
}
```

</details>