Samenvatting Analyses 2 HBO Informatica 1e jaar 2021/2022

Analysis 2: Foundations of Modeling 2


1.1 Functions in mathematics

A relation is a relationship between sets of values.
A mathematical relation is an association of two objects, based on some property possessed by them.

In mathematics, relations between two sets can be expressed in:
 Roster form
 Set-builder form
 Arrow diagram

Roster form
 Relation R is represented as a set of ordered pairs.
Let A = {1, 5} and B = {2, 5, 10}
 Let R be the relation between sets A and B such that “a is less than b” :
R = { (1,2), (1,5), (1,10), (5,10) }
Note: A x B = {(1,2),(1,5),(1,10), (5,2),(5,5),(5,10)}, hence R ⊆ A x B.

Set-builder form
 The same example and its relation can be written in set-builder form:
Let A = {1, 5} and B = {2, 5, 10}
 Then the relation “a is less than b” can be written as:
R = {(a, b) : a ∈ A, b ∈ B, a is less than b}, or
R = {(a, b) : a ∈ A, b ∈ B, a < b}

Problem 1: expressing a real-life relation
Let A = {Cow, Chicken, Crocus} and B = {Plant, Animal, Bird, Mammal}. Let the relation R from A to B
be defined as “is a”.
(a) Express R in set-builder notation.

(b) Express R in roster notation.
R = {(Cow, Animal), (Cow, Mammal), (Chicken, Animal), (Chicken, Bird), (Crocus, Plant)}

(c) Express R in an arrow diagram.




Plant
Cow
Animal
Chicken
Bird
Crocus
Mammal

,A function is a special kind of relation.
 If f is a mapping from A to B,
we express it as f: A → B,
and read it as “f is a function from A to B”.

A function consists of three parts:
 input
 relationship
 output
A E
B F
Function as a relation:
C
 Every function is special kind of relation. G
D
 However, not every relation is a function!
 f: A → B is the relation that to each a ∈ A associates f(a) ∈ B.

What makes a relation a function?
 Every element of A must have an image in B. A E A
E
 No element of A can have more than one image. B F B
F
C G C
G
f(x) = 2x H D
f = function name
x = input
2x = output

Input Relationship Output
0 *2 0
1 *2 2
7 *2 14

Domain: what can go into function.
Codomain: what may possibly come out of a function.
Range: what actually comes out of a function.
1
1 4
9
Evaluation of functions 2 16
25 Range
Given the function: f(x) = x**2 + 2x 3
Evaluate f(0) = 0**2 + 2 * 0 = 0 4
Evaluate f(5) = 5**2 + 2 * 5 = 35 17
5 23
19


Domain Codomain

, Recognize basic function forms