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

Example 64 On sets of persons, the “mother-of” relation is an non-symmetric, non-reflexive
relation.
Example 65 On sets of persons, the “ancestor-of” relation is a partial order that is not
linear.
Functions (as special relations)
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Definition 66 f ⊆ X × Y , is called a partial function, iff for all x ∈ X there is at most
one y ∈ Y with 〈x, y〉 ∈ f.
Notation 67 f : X ⇀ Y ; x ↦→ y if 〈x, y〉 ∈ f (arrow notation)
call X the domain (write dom(f)), and Y the codomain (codom(f)) (come with f)
Notation 68 f(x) = y instead of 〈x, y〉 ∈ f (function application)
Definition 69 We call a partial function f : X ⇀ Y undefined at x ∈ X, iff 〈x, y〉 ∈ f for
all y ∈ Y . (write f(x) = ⊥)
Definition 70 If f : X ⇀ Y is a total relation, we call f a total function and write f : X →
Y . (∀x ∈ X.∃ 1 y ∈ Y .〈x, y〉 ∈ f)
Notation 71 f : x ↦→ y if 〈x, y〉 ∈ f (arrow notation)
: this probably does not conform to your intuition about functions. Do not
worry, just think of them as two different things they will come together over time.
(In this course we will use “function” as defined here!)
Function Spaces
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Definition 72 Given sets A and B We will call the set A → B (A ⇀ B) of all (partial)
functions from A to B the (partial) function space from A to B.
Example 73 Let B := {0, 1} be a two-element set, then
B → B = {{〈0, 0〉, 〈1, 0〉}, {〈0, 1〉, 〈1, 1〉}, {〈0, 1〉, 〈1, 0〉}, {〈0, 0〉, 〈1, 1〉}}
B ⇀ B = B → B ∪ {∅, {〈0, 0〉}, {〈0, 1〉}, {〈1, 0〉}, {〈1, 1〉}}
as we can see, all of these functions are finite (as relations)
Lambda-Notation for Functions
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Problem: It is common mathematical practice to write things like fa(x) = ax 2 +
3x + 5, meaning e.g. that we have a collection {fa | a ∈ A} of functions.
(is a an argument or jut a “parameter”?)
Definition 74 To make the role of arguments extremely clear, we write functions in λnotation.
For f = {〈x, E〉 | x ∈ X}, where E is an expression, we write λx ∈ X.E.
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