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Jan Meyer
2026-03-07 09:46:32 +01:00
parent 86dc0e0b9c
commit 3750d6fd55
4 changed files with 60 additions and 14 deletions
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.obsidian/graph.json vendored
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DAS/Relations.md Normal file
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## Types of Relations
| Relation | Explanation | Example |
| ---------------- | :-------------------------------------------------------------------------------------------------------------------- | ------------------------- |
| *transitive*<br> | "chain reaction", a information about $a$ in relation to $c$ can be inferred from the relations $a -> b$ and $b -> c$ | $a < b, b < c => a < c$ |
| *reflexive* | every element is related to itself with the given relation | $a <= a, 5 = 5$ |
| *anti-reflexive* | every element is *NOT* related to itself in the given relation | $a < a$ |
| *symmetric* | the given relation work both ways | $a = b => b = a$ |
| *antisymmetric* | the given relation only works both ways if $a$ and $b$ are the same | $a <= b, b <= a => a = b$ |
## Equivalence Relations
A relation $R$ is called _equivalence relation_ when it is _transitive, reflexive and symmetric_.
### Example:
**Question:** How many equivalence classes are there for the given equivalence relation?
$$
& ~ "on" {0, 1, 2, 3}^(2) \
& "defined by" (x_1, y_1) ~ (x_2, y_2) <==> x_1 + y_1 = x_2 + y_2
$$
> [!INFO]
> Meaning:
> The pairs $(x_1, y_1)$ and $(x_2, y_2)$ are equivalent to each other when the components of the pair added up have the same result.
Solving:
- Smallest possible sum: $(0 + 0) = 0$
- Biggest possible sum: $(3 + 3) = 6$
- All possible sums: $0, 1, 2, 3, 4, 5, 6$
Each possible sum creates it's own equivalence class. So there are $7$ equivalence classes.
> [!NOTE]
> All equivalence classes:
> $[0]_(~) = {(0, 0)}$
> $[1]_(~) = {(0, 1), (1, 0)}$
> $[2]_(~) = {(0, 2), (1, 1), (2, 0)}$
>$[3]_(~) = {(0, 3), (1, 2), (2, 1), (3, 0)}$
>$[4]_(~) = {(1, 3), (2, 2), (3, 1)}$
> $[5]_(~) = {(2, 3), (3, 2)}$
> $[6]_(~) = {(3, 3)}$

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DAS/Set Theory.md
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@@ -84,12 +84,3 @@ A := {1, 2, 3} \
cal(P)(A) = {emptyset, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}
$$
## Relations
| Relation | Explanation | Example |
| ---------------- | :-------------------------------------------------------------------------------------------------------------------- | ------------------------- |
| *transitive*<br> | "chain reaction", a information about $a$ in relation to $c$ can be inferred from the relations $a -> b$ and $b -> c$ | $a < b, b < c => a < c$ |
| *reflexive* | every element is related to itself with the given relation | $a <= a, 5 = 5$ |
| *anti-reflexive* | every element is *NOT* related to itself in the given relation | $a < a$ |
| *symmetric* | the given relation work both ways | $a = b => b = a$ |
| *antisymmetric* | the given relation only works both ways if $a$ and $b$ are the same | $a <= b, b <= a => a = b$ |