54 lines
2.6 KiB
Markdown
54 lines
2.6 KiB
Markdown
## Types of Relations
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| Relation | Explanation | Example |
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| ---------------- | :-------------------------------------------------------------------------------------------------------------------- | ------------------------- |
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| *transitive*<br> | "chain reaction", a information about $a$ in relation to $c$ can be inferred from the relations $a -> b$ and $b -> c$ | $a < b, b < c => a < c$ |
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| *reflexive* | every element is related to itself with the given relation | $a <= a, 5 = 5$ |
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| *anti-reflexive* | every element is *NOT* related to itself in the given relation | $a < a$ |
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| *symmetric* | the given relation work both ways | $a = b => b = a$ |
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| *antisymmetric* | the given relation only works both ways if $a$ and $b$ are the same | $a <= b, b <= a => a = b$ |
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## Equivalence Relations
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A relation $R$ is called _equivalence relation_ when it is _transitive, reflexive and symmetric_.
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### Example:
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**Question:** How many equivalence classes are there for the given equivalence relation?
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$$
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& ~ "on" {0, 1, 2, 3}^(2) \
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& "defined by" (x_1, y_1) ~ (x_2, y_2) <==> x_1 + y_1 = x_2 + y_2
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$$
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> [!INFO]
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> Meaning:
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> 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.
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Solving:
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- Smallest possible sum: $(0 + 0) = 0$
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- Biggest possible sum: $(3 + 3) = 6$
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- All possible sums: $0, 1, 2, 3, 4, 5, 6$
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Each possible sum creates it's own equivalence class. So there are $7$ equivalence classes.
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> [!NOTE]
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> All equivalence classes:
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> $[0]_(~) = {(0, 0)}$
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> $[1]_(~) = {(0, 1), (1, 0)}$
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> $[2]_(~) = {(0, 2), (1, 1), (2, 0)}$
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>$[3]_(~) = {(0, 3), (1, 2), (2, 1), (3, 0)}$
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>$[4]_(~) = {(1, 3), (2, 2), (3, 1)}$
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> $[5]_(~) = {(2, 3), (3, 2)}$
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> $[6]_(~) = {(3, 3)}$
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## Binary Relation
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A binary relation is a relation $R$ between _exactly two_ elements $a in R$ and $b in R$. An example for a binary relation is $a <= b$
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## Converse Relation
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$C^top$ or $C^(-1)$ is the
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## Orders
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An **Order** is a mathematical way to sort, rank or compare elements within a set, where some elements come "before" and "after" others.
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A _binary relation_ is called an order if it is...
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- [?] a *reflexive relation*
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- [?] a *antisymmetric relation*
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- [?] a *transitive relation*
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