uni_notes/DAS/Relations.md

Types of Relations

Relation	Explanation	Example
transitive	"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)}

Binary Relation

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

Orders

An Order is a mathematical way to sort, rank or compare elements within a set, where some elements come "before" and "after" others.

A binary relation is called an order if it is...

• a reflexive relation
• a antisymmetric relation