A function is a mapping from one set , the domain, to another
set, the co-domain.
Each element of the domain must have an image in the codomain.
We write f: X→Y.
Illustrating functions:
1. Arrow diagrams |
domain:
co-domain:
range of f:
inverse image of 0, 1, 2: |
2. Function machine
3. Using functional notation
4. Graphing
5. Listing all sets of ordered pairs
Two functions f and g are equal if they have the same domains and co-domains and
for every x
in the domain, f(x) = g(x).
Examples of functions:
1. A sequence : 1, 1/2, 1/4, ... is a function from Z+ to R.
2. the cardinality of a subset is a function from P (X) to non- negative
integers .
ex. X = { a, b, c } P (X) =
3. a 2-input circuit is a function from {0,1}×{0,1} to {0,1}.
This circuit is equivalent to what statement ?
This is an example of a Boolean function.
4. Hamming distance is a function from Σn × Σn to Znonnegative
where Σ = { 0, 1 }
ex: Σ3 = {
One-to-one (Injective)
|
To show a function is injective, show that if f(x1) = f(x2), then x1 = x2.
To show a function is NOT injective, give a counterexample, i.e., show a case
where
f(x1) = f(x2) but x1 ≠ x2. |
ex 1. Is the function f: R to R by f(x) = −5x + 1 is one-to-one? Show why.
ex 2. Is the function g: R to R by g(x) = x2 + 2 is one-to-one? Show why.
ex 3. Is the function h: R to R by h(x) = x3 is one-to-one? Show
why.
Onto (Surjective) Every element in the co-domain is an image of some element in
the domain.
ex. 1: Show whether the function f: R to R by f(x) = x2 + 2 is onto.
ex. 2: Show whether the function g: R to R by g(x) = 3x + 1 is onto.
ex. 3: Show whether the function h: R to R by h(x) = x2 is onto.
One-to-one and onto (Bijective or one-to-one corresponding)
Is the function f: R to R by f(x) = 5x + 1 bijective?
|
If f is one-to-one and onto, f has an inverse function, f-1. |
Hash functions - used to de termine where to store data and where to find data
that has been
stored.
“Hash” the data by performing some mathematical operation on the key . The
resulting number
tells where to store the data.
ex. Store student records in an array indexed 0..999.
Hash the student ID as follows
H(ID) = ID mod 1000
|
 |
| What's the big advantage to a hash function? |
 |
Suppose we use the scheme
h(Student) = last 2 digits of the ID
as a hash function to store Student data in an array, i.e,
index = h(Student)
Where are Student records of students with IDs shown be low stored ?
| 091277817 |
 |
| 112376149 |
| 116724516 |
| |
Can there be a Problem?
Pigeonhole Principle:
A function from one finite set to a smaller finite set cannot be one-toone.
There must be at least 2 elements in the domain that have the
same image in the co-domain.
ex: In this room, must there be at least 2 people of the same sex?
ex.: How many people would have to use the pigeonhole principle to guarantee
that at least 2
people shared the same birth month?
Generalized Pigeonhole Principle:
For function f: X
Y where X and Y are finite sets and
N(X) > k * N(Y) for some k ∈Z+,
then at least k + 1 elements of X have the same image under f.
ex: At least _____ people in the room have the same birth month.
Y has cardinality? Largest k = ?
ex. At least _____ people in the room have the same sex.
ex. (As sume there are only 4 hair colors: blond, black, brunette, red hair)
At least _____ people in the room have the same hair color.
ex. (Assume there are only 3 majors: CS, MA, EG)
At least _____ people in the room have the same major.
Composition of Functions: Applying a function to the result of applying a
function
 |
Let g: Z Z and f: Z
Z |
| |
 |
| ex. Let f(x) = 2x and g(x) = x2 + 3 |
Let g: Z Z and f: Z
Z |
| Find (f ° g) (5) |
|
| Find (g ° f) (5) |
|
| Is there a function h(x) = (f ° g) (x) ?
|
Is it 1 to 1? Onto? |
| Is there a function j(x) = (g ° f) (x) ? |
|
|
If functions f and g are both
one-to-one, f ° g is one-to-one.
If f and g are both onto, f ° g is onto.
If f and g are both one-to-one corresponding, f ° g is one-to-one
corresponding. |
x ∈ domain 
f(x) = y.