Introduction
Functions
Discrete Mathematics
Andrei Bulatov
Discrete Mathematics  Functions
162
Previous Lecture
Properties of binary relations
Equivalence relations and partitions
Partial and total orders
Diagrams of orders
square4
reflexivity
square4
symmetricity
square4
transitivity
square4
antisymmetricity
Discrete Mathematics  Functions
163
Functions
In many instances we assign to each element of a set a particular
element of a second set.
For example , assign rooms to people in a hotel
Or
we may assign a grade to each student from a class
What we get is a set of pairs
(Person, Door)
or
(Student, Grade),
that is a relation, but a very particular one
Discrete Mathematics  Functions
164
Functions (cntd)
A relation
R
from
A
to
B
is called
a
function
from
A
to
B, if for
every
a
∈
A
there is exactly one
b
∈
B
such that
(a,b)
∈
R. (Also
mappings
,
transformations
)
Adams
Chou
Goodfriend
Rodriguez
Stevens
A
B
C
D
F
We use
f,g,h
to denote functions
f: A
→
B
f(a) = b
f(Rodriguez) = A
Discrete Mathematics  Functions
165
Example
Consider the function from the set People to People:
f(a) = b
if
b
is the father of
a.
James VI of Scotland
Charles I
Charles II
Mary
James II
Elizabeth
Anne
Henry
Henrietta
William II
William III
Mary II
Anne
Henry Frederick
Elizabeth
Frederick V
Sophia (Queen Anne)
Rupert
Charles Louis
Ernest Augustus
George I
(http://www.royal.gov.uk)
Discrete Mathematics  Functions
166
Domain and Codomain
Let
f: A
→
B
be a function from
A
to
B.
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 Spring '10
 AndreiBulatov
 Binary relation, Bijection, Adams Chou Goodfriend Rodriguez Stevens, Adams Chou Goodfriend

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