General Computer Science 320201 GenCS I & II Lecture ... - Kwarc

2.2.4 Relations and Functions
Now we will take a closer look at two very fundamental notions in mathematics: functions and
relations. Intuitively, functions are mathematical objects that take arguments (as input) and
return a result (as output), whereas relations are objects that take arguments and state whether
they are related.
We have alread encountered functions and relations as set operations — e.g. the elementhood
relation ∈ which relates a set to its elements or the powerset function that takes a set and produces
another (its powerset).
Relations
Definition 56 R ⊆ A × B is a (binary) relation between A and B.
Definition 57 If A = B then R is called a relation on A.
Definition 58 R ⊆ A × B is called total iff ∀x ∈ A.∃y ∈ B.〈x, y〉 ∈ R.
Definition 59 R −1 := {〈y, x〉 | 〈x, y〉 ∈ R} is the converse relation of R.
Note: R −1 ⊆ B × A.
The composition of R ⊆ A × B and S ⊆ B × C is defined as S ◦ R :=
{〈a, c〉 ∈ (A × C) | ∃b ∈ B.〈a, b〉 ∈ R ∧ 〈b, c〉 ∈ S}
Example 60 relation ⊆, =, has color
Note: we do not really need ternary, quaternary, . . . relations
Idea: Consider A × B × C as A × (B × C) and 〈a, b, c〉 as 〈a, 〈b, c〉〉
we can (and often will) see 〈a, b, c〉 as 〈a, 〈b, c〉〉 different representations of the same
object.
c○: Michael Kohlhase 50
We will need certain classes of relations in following, so we introduce the necessary abstract
properties of relations.
Properties of binary Relations
Definition 61 A relation R ⊆ A × A is called
reflexive on A, iff ∀a ∈ A.〈a, a〉 ∈ R
symmetric on A, iff ∀a, b ∈ A.〈a, b〉 ∈ R ⇒ 〈b, a〉 ∈ R
antisymmetric on A, iff ∀a, b ∈ A.(〈a, b〉 ∈ R ∧ 〈b, a〉 ∈ R) ⇒ a = b
transitive on A, iff ∀a, b, c ∈ A.(〈a, b〉 ∈ R ∧ 〈b, c〉 ∈ R) ⇒ 〈a, c〉 ∈ R
equivalence relation on A, iff R is reflexive, symmetric, and transitive
partial order on A, iff R is reflexive, antisymmetric, and transitive on A.
a linear order on A, iff R is transitive and for all x, y ∈ A with x = y either 〈x, y〉 ∈ R or
〈y, x〉 ∈ R
Example 62 The equality relation is an equivalence relation on any set.
Example 63 The ≤ relation is a linear order on N (all elements are comparable)
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