DISCRETE MATHEMATICAL
FUNCTIONS
Suppose we have:
And I ask you to describe the yellow function.
What’s a function?
Notation: f: RR, f(x) = -(1/2)x - 25
domain
co-domain
f(x) = -(1/2)x - 25
FUNCTIONS
Definition: a function f : A  B is a subset of AxB
where  a  A, ! b  B and <a,b>  f.
FUNCTIONS
A collection of
points!
Definition: a function f : A  B is a subset of AxB
where  a  A, ! b  B and <a,b>  f.
A point!
B
B
A
A
FUNCTIONS
A = {Michael, Tito, Janet, Cindy, Bobby}
B = {Katherine Scruse, Carol Brady, Mother Teresa}
Let f: A  B be defined as f(a) = mother(a).
Michael
Tito
Janet
Cindy
Bobby
Katherine
Scruse
Carol Brady
Mother Teresa
FUNCTIONS - IMAGE & PREIMAGE
What about
the range?
For any set S  A, image(S) = {b : a  S, f(a) = b}
image(S) = f(S)
So, image({Michael, Tito}) = {Katherine Scruse}
image(A) = B - {Mother Teresa}
Michael
Tito
Janet
Cindy
Bobby
Katherine
Scruse
Carol Brady
Mother Teresa
Some say it
means
codomain,
others say,
image. Since
it’s ambiguous,
we don’t use it
at all.
FUNCTIONS - IMAGE & PREIMAGE
preimage(S) = f-1(S)
For any S  B, preimage(S) = {a: b  S, f(a) = b}
So, preimage({Carol Brady}) = {Cindy, Bobby}
preimage(B) = A
Michael
Tito
Janet
Cindy
Bobby
Katherine
Scruse
Carol Brady
Mother Teresa
Every b  B has
at most 1
preimage.
FUNCTIONS - INJECTION
A function f: A  B is one-to-one (injective, an
injection) if a,b,c, (f(a) = b  f(c) = b)  a = c
Not one-to-one
Michael
Tito
Janet
Cindy
Bobby
Katherine
Scruse
Carol Brady
Mother Teresa
FUNCTIONS - SURJECTION
Every b  B has
at least 1
preimage.
A function f: A  B is onto (surjective, a surjection)
if b  B, a  A f(a) = b
Not onto
Michael
Tito
Janet
Cindy
Bobby
Katherine
Scruse
Carol Brady
Mother Teresa
FUNCTIONS - BIJECTION
A function f: A  B is bijective if it is one-to-one
and onto.
Every b  B has
exactly 1
preimage.
Isaak
Bri
Lynette
Aidan
Evan
Cinda
Dee
Deb
Katrina
Dawn
An important
implication of this
characteristic:
The preimage (f-1)
is a function!
FUNCTIONS - EXAMPLES
Suppose f: R+  R+, f(x) = x2.
Is f one-to-one? yes
Is f onto? yes
Is f bijective?
yes
FUNCTIONS - EXAMPLES
Suppose f: R  R+, f(x) = x2.
Is f one-to-one?
Is f onto? yes
Is f bijective?
no
no
FUNCTIONS - EXAMPLES
Suppose f: R  R, f(x) = x2.
Is f one-to-one?
Is f onto?
no
no
Is f bijective? no
FUNCTIONS - COMPOSITION
Let f:AB, and g:BC be functions. Then the
composition of f and g is:
(g o f)(x) = g(f(x))