Discussion #29
Functions
Discussion #29
1/20
Topics
•
•
•
•
•
•
•
•
Function Definition
Notation
Partial Functions
Restrictions
Overloading
Composition
Injections, surjections, bijections
Inverses
Discussion #29
2/20
Function Definition
• A function is a special kind of binary
relation.
• A binary relation f  A  B is a function if
for each a  A there is a unique b  B
y
1
α
2
β
3
γ
x
Discussion #29
3/20
NOT Functions
1
α
2
β
3
γ
f = {(1, α), (2, β)}
“For each” violated
y
Some x’s do not have
corresponding y’s
x
Discussion #29
4/20
NOT Functions
1
α
2
β
3
γ
f = {(1, α), (2, β), (3, β), (3, γ)}
uniqueness violated for 3 
appears twice
y
Uniqueness violated for some x’s
x
Discussion #29
5/20
Functions
with N-Dimensional Domains
An (n+1)-ary relation f  A1  A2  … 
An  B is a function if for each < a1, a2,
…, an>  A1  A2  …  An there is a
unique b  B.
Discussion #29
<1,1>
α
<1,2>
β
<1,3>
γ
6/20
Notation for Functions
• We can use various notation for functions: for
f = {(1, α),(2, β),(3, β)}
Notation
(x, y)  f
f : x→y
y = f(x)
Example
(2, β)  f
f : 2→β
β = f(2)
• In the notation, x is the argument or preimage
and y is the image. We can also have the
image of a set of arguments.
• For functions with n-ary domains, use <x0, x1,
…, xn> in place of x.
Discussion #29
7/20
Function Domain and Range
• f:A→B
– A is the domain space
• same as the domain (since all elements participate)
• dom f, dom(f), or domain(f)
– B is the range space
• may or may not be the same as the range, which is:
– {y | x(y=f(x))}
– All rhs values in pairs (all that get “hit”)
– Bf
• ran f, ran(f), range(f)
• f : D1  D2  …  Dn → Z
• f : Dn → Z (when all domains are the same)
Discussion #29
8/20
Partial Functions
Remove the requirement that each a  A must
participate. Retain the uniqueness requirement.
Partial Function:
<1,1>
α
<1,2>
β
<1,3>
γ
NOT a Partial Function:
<1,1>
α
<1,2>
β
<1,3>
Discussion #29
γ
Partial Function: (A
Total Function is also a
Partial Function.)
<1,1>
α
<1,2>
β
<1,3>
γ
f = {(<1,2>, β),(<1,3>, β),(<1,3>, γ)}
<1,3> not unique
9/20
Special Functions
• Identity Function
– IA : A → A
– IA = {(x, x) | x  A}
• Constant Function
–C:A→B
– C = {(x, c) | x  A  c  B }
– Often A and B are the same
• C:A→A
• C= {(x, c) | x  A  c  A}
Discussion #29
10/20
Restrictions
It is common to restrict the domain space of
partial functions to be the actual domain.
– log(x) is partial for R → R but total for R>0 → R
– For “number of pregnancies” we may restrict the domain to mothers
John
1
Mary
2
Sue
Ken
3
– Sometimes we restrict for other reasons, e.g. mothers over 40
John
1
Mary
2
Sue
Ken
Discussion #29
3
11/20
Overloading
• Overloading is like overloading in Java.
• The same name can be used for different
functions depending on the domain.
• Examples
a–b
• Means number subtraction if a and b are numbers
• Means set subtraction if a and b are sets
a+b
• Implemented differently for integers and reals
• Could potentially be overridden and implemented by
a programmer for very long integers
Discussion #29
12/20
Composition of Functions
a
f
1
2
b
3
g(2) = α
g(f(a)) = α
g°f(a) = α
f(b) = 2
α
β
4
c
f(a) = 2
g
g(2) = α
g(f(b)) = α
• Composition is written “°”
• Range space of f =
domain space of g
g°f(b) = α
f(c) = 4
g(4) = β
g(f(c)) = β
g°f(c) = β
Discussion #29
13/20
Injection
Injection: “one-to-one” or “1-1”
– xy(f(x) = f(y)  x = y)
– For f : A → B, the elements in B are “hit” at most once
Injective
NOT Injective
1
a
1
b
b
2
3
2
c
d
3
c
d
y
y
x
Discussion #29
a
x
14/20
Surjection
Surjection: “onto”
– yx(y = f(x))
– For f : A → B, the elements in B are all “hit” at least once
Surjective
NOT Surjective
1
2
3
4
a
1
2
b
3
4
c
b
c
y
y
x
Discussion #29
a
{
not
“hit”
x
15/20
Bijection
Bijection: “one-to-one and onto” or “1-1 correspondence”
– xy(f(x) = f(y)  x = y)  yx(y = f(x))
– For f : A → B, every B element is “hit” once and only once
Bijective
NOT Bijective
1
2
a
1
2
b
3
3
4
c
NOT injective
a
b
c
y
y
x
Discussion #29
NOT Surjective
x
16/20
Notes on Bijection
1. |A| = |B|
–
–
An “extra” B cannot be “hit” (not a surjection)
An “extra” A requires that at least one B must be
“hit” twice (not an injection)
2. If f is a bijection, swapping the elements of the
ordered pairs is a function
–
–
–
–
Called the inverse
Denoted f-1
Is also a bijection
f-1(f(x)) is the identity function, i.e. f-1(f(x)) = x.
Discussion #29
17/20
Notes on Bijection (continued …)
3. The inverse of an injection is a partial function.
If f : A →B is an injection, then f-1 is a partial function
f
1
f-1
a
a
b
b
2
c
d
3
c
d
1
2
3
4. Restricting the range space of an injective
function to the range yields a bijection
Remove b
Discussion #29
18/20
Which is bigger? N or R[0..1]?
Assume |N| = |R[0..1]|, then there exists a
bijection:
1 0.34234…
2 0.34987…
diagonalization
3 0.00040…
But now there exists a number in R[0..1] such that
d1 = not 3, d2 = not 4, d3 = not 0, … . Hence, not
surjective and thus not bijective.
Discussion #29
19/20
Which is bigger? N or Z?
f(x) =
g(y) =
{
{
x odd: (x+1)/−2
x even: x/2
y negative: −2x−1
y positive: 2x
x
0
1
2
3
4
y
0
−1
1
−2
2
Since g = f−1, there is a bijection from N to Z and thus |N| = |Z|.
Countable if same cardinality as some subset of N.
Alternatively, a set S is countable if there exists an injective
function from S to N.
Discussion #29
20/20