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
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
