Download Functions Defined on General Sets

CSE 215: Foundations of
Computer Science
Unit 16:
Functions Defined
on General Sets
Kevin McDonnell
Stony Brook University – CSE 215
1
Functions
• A function f from a set X to a set Y, denoted : → , is a
relation from X, the domain, to Y, the co-domain, that
satisfies two properties:
1. Every element in X is related to some element in Y
2. No element in X is related to more than one element in Y
• For any element ∈ , there is a unique element ∈
such that ( ) =
• The range of f (also called the image of X under f )
∈
=
for some ∈ }
=
• The inverse image of y =
∈
( )= }
Kevin McDonnell
Stony Brook University – CSE 215
2
Arrow Diagrams
• If X and Y are finite sets, you can define a function f from X
to Y by drawing an arrow diagram
• You make a list of elements in X and a list of elements in Y,
and draw an arrow from each element in X to the
corresponding element in Y, as shown below
Kevin McDonnell
Stony Brook University – CSE 215
3
Arrow Diagrams
• This arrow diagram does define a function because
1. Every element of X has an arrow coming out of it.
2. No element of X has two arrows coming out of it that
point to two different elements of Y.
Kevin McDonnell
Stony Brook University – CSE 215
4
Arrow Diagrams
• For each arrow diagram below, indicate whether the
diagram depicts a function
No
Kevin McDonnell
No
Stony Brook University – CSE 215
Yes
5
Example: Arrow Diagrams
• Let X = {a, b, c} and Y = {1, 2, 3, 4}. Define a function f from
X to Y by the arrow diagram below.
a. Write the domain and co-domain of f.
• Answer: domain: { , , } co-domain: {1, 2, 3, 4}
b. Find f(a), f(b), and f(c).
• Answer:
= 2,
= 4,
=2
c. What is the range of f?
• Answer: {2,4}
Kevin McDonnell
Stony Brook University – CSE 215
6
Example: Arrow Diagrams
• Let X = {a, b, c} and Y = {1, 2, 3, 4}. Define a function f from
X to Y by the arrow diagram below.
d. Is c an inverse image of 2? Is b an inverse image of 3?
• Answer: Yes, No
e. Find the inverse images of 2, 4, and 1.
• Answer: , ,
,∅
f. Represent f as a set of ordered pairs.
• Answer: { , 2 , , 4 , , 2 }
Kevin McDonnell
Stony Brook University – CSE 215
7
Arrow Diagrams
• For this function there are no arrows pointing to the 1 or
the 3.
• This illustrates the fact that although each element of the
domain of a function must have an arrow pointing out from
it, there can be elements of the co-domain to which no
arrows point.
• Note also that there are two
arrows pointing to the 2:
one coming from a and the
other from c.
Kevin McDonnell
Stony Brook University – CSE 215
8
Function Equality
• If : → and : → are functions, then = if, and
only if, ( ) = ( ) for all ∈ .
• Example: Let J3 = {0, 1, 2}, and define functions f and g from
=
+ + 1 mod 3
J3 to J3 as follows: For all x in J3,
and #
= + 2 mod 3
• Does = #?
• Yes, the table of values shows that f(x) = g(x) for all x in J3.
x $ % = %& + % + ' ()* + , % = % + &
0
1
2
Kevin McDonnell
1 -./ 3 = 1
3 -./ 3 = 0
7 -./ 3 = 1
&
()* +
4 -./ 3 = 1
9 -./ 3 = 0
16 -./ 3 = 1
Stony Brook University – CSE 215
9
Example: Function Equality
• Let F: ℝ → ℝ and G: ℝ → ℝ be functions.
• Define new functions F + G: ℝ → ℝ and G + F: ℝ → ℝ as
follows: for all x ∈ ℝ,
+
=
+ ( ) and
+
=
+
.
• Does F + G = G + F ?
• Yes. For all real numbers x:
+
=
+ ( )
by definition of +
+ ( )
by the commutative law
=
=
+
by definition of +
Kevin McDonnell
Stony Brook University – CSE 215
10
The Identity Function
• Given a set X, define a function IX from X to X by 56 ( ) =
for all x in X.
• The function IX is called the identity function on X
because it sends each element of X to the element that is
identical to it.
• Thus the identity function can be pictured as a machine
that sends each piece of input directly to the output chute
without changing it in any way.
Kevin McDonnell
Stony Brook University – CSE 215
11
Logarithmic Functions
• Let x be a positive real number with ≠ 1. For each
positive real number x, the logarithm with base b of x,
written log : , is the exponent to which b must be raised to
obtain x. Symbolically, log : =
⇔ <= .
• The logarithm function with base b is the function from
ℝ= to ℝ that takes each positive real number x to log : .
• Examples: evaluate each of the following:
a. log > 9 = 2
b. log
?
= −1
c. log?@ 1 = 0
d. log 2A = m
e. 2
BCDE A
Kevin McDonnell
->0 = m
This is equivalent to saying log - = log -.
Then by definition, 2BCDE A = -.
Stony Brook University – CSE 215
12
Sec. 7.1, p. 303: #23
• If b and y are positive real numbers such that log :
what is log H
?
= 3,
I
= 3, then
• If log :
•
>
=
=
=
=
=
>
by definition of log.
?
:KL
?
H
IL
?
H L
I
? M>
:
So
• By definition of log, log
Kevin McDonnell
=
H
I
? M>
:
= −3
Stony Brook University – CSE 215
13
Function Defined on a Power Set
• Define a function : N
each ∈ N , , ,
Kevin McDonnell
, ,
→ ℤPQPPRS as follows: for
= the number of elements in X.
Stony Brook University – CSE 215
15
Function for a Sequence
• Suppose we are given this sequence:
? ?
? ?
M? P
1, − , , − , , … ,
,…
P=?
>
T U
• We can think of this sequence as a function from the
nonnegative integers to real numbers as follows:
?
0 → 1, 1 → − , 2 →
?
,
>
3 →
?
− ,
T
4 →
?
, …, W
U
→
M? P
P=?
• This can be written formally as a function:
: ℕ → ℝ, for each integer W ≥ 0,
W =
M? Z
(P=?)
#: ℤ= → ℝ, for each integer W ≥ 1, # W =
or as
M? Z[H
P
• Many functions can be used to define a given sequence
Kevin McDonnell
Stony Brook University – CSE 215
16
Function for a Cartesian Product
• Define functions \: ℝ × ℝ → ℝ and ^: ℝ × ℝ → ℝ × ℝ as
follows. For all ordered pairs , of integers,
and ^ , = (− , ).
\ , =
• M is called the multiplication function
• R is called the reflection function
• Evaluate each of the following:
a. \ −1, −1 = 1
b. \
? ?
,
=
?
T
c. ^ −2,5 = 2,5
d. ^ 3, −4 = −3, −4
Kevin McDonnell
Stony Brook University – CSE 215
17
The Hamming Distance Function
• Let `P be the set of all strings of 0’s and 1’s of length n.
a: `P × `P → ℤPQPPRS
• For each pair of strings (b, c) ∈ `P × `P , a(b, c) = the
number of positions in which s and t differ
• For W = 5, a 11111, 00000 = 5
a(10101, 01101) = 2
a(01010, 11100) = 3
• The Hamming distance function is used in certain error
correction and error detection algorithms
Kevin McDonnell
Stony Brook University – CSE 215
18
Well-defined Functions
• Sometimes what appears to be a function defined by a rule
is not really a function at all
• Consider this sentence: “Define a function
f : ℝ → ℝ by specifying that for all real numbers x, ( ) is
= 1.”
the real number y such that +
• There are two problems with this purported function. For
almost all values of x, either
1. there is no y that satisfies the given equation (try = 2),
or
2. there are two different values of y that satisfy the
equation (try = 0).
Kevin McDonnell
Stony Brook University – CSE 215
19
Example: Non-well-defined Function
• Suppose you read that a function f : ℚ → ℤ is to be defined
A
by the formula
= - for all integers m and n with
P
W ≠ 0.
• That is, the integer associated by f to the
A
number
P
is m.
• Is f a well-defined function?
?
• Answer: No. Consider and
>
. Now if f were
e
a function, then
the definition of a function would imply that
• But
?
= 1 and
>
e
?
=
>
e
.
= 3 and 1 ≠ 3. So f is not a
function.
Kevin McDonnell
Stony Brook University – CSE 215
20
Functions Acting on Sets
• If : → is a function and f ⊆ and h ⊆ , then
f =
∈
= ( for some in f and
M?
h =
∈
( ) ∈ h}
• (f) is called the image of A, and M? h is called the
inverse image of C.
Kevin McDonnell
Stony Brook University – CSE 215
21
Example: Functions Acting on Sets
• Let X = {1, 2, 3, 4} and Y = {a, b, c, d, e}, and define
F : X → Y by the following arrow diagram:
Let A = {1, 4}, C = {a, b},
and D = {c, e}.
• Find each of the following:
• f = {b}
•
= {a, b, d}
• M? h = {1, 2, 4}
• M? k = Ø
Kevin McDonnell
Stony Brook University – CSE 215
22
Example: Functions Acting on Sets
• Let X and Y be sets, let F: X → Y be a function. Prove that if
f ⊆ and l ⊆ , then (f ∪ l) ⊆ (f) ∪ (l)
• Proof. Suppose ∈ f ∪ l .
• This means = ( ) for some ∈ f ∪ l.
• By definition of union, ∈ f or ∈ l.
• Case 1, ∈ f. In this case, = ( ) for some in A.
• Hence, ∈ f , and so by definition of union,
∈ (f) ∪ (l).
• Case 2, ∉ f, which means ∈ l. In this case, = ( ) for
some in B.
• Hence, ∈ l , and so by definition of union,
∈ (f) ∪ (l). Thus, in either case, ∈ (f) ∪ (l).
Kevin McDonnell
Stony Brook University – CSE 215
23
Sec. 7.1, p. 304: #36
• True or false: for sets A and B, which are subsets of a set X,
and for a function F from X to another set Y,
f ∩ (l) ⊆
f∩l ?
• This is false.
• Counterexample: Let = {1,2,3}, = { , }, f = 1,2 ,
l = {1,3}.
• Let
1 = { },
2 = { },
3 ={ }
• Let f = { , }, l = ,
• Then f ∩ l = { , }
• f ∩ l = {1}, so f ∩ l = { }
• Thus, f ∩ l ⊈ f ∩ l
Kevin McDonnell
Stony Brook University – CSE 215
25
Sec. 7.1, p. 304: #38
• True or false: for sets C and D, which are subsets of set Y,
and for a function F from X to Y, if h ⊆ k, then
M?
h ⊆ M? (k)?
• This is true.
• Suppose ∈ M? h .
• Then
∈ h.
• Since h ⊆ k,
∈ k also.
• This means that ∈ M? k .
• Therefore, M? h ⊆ M? k .
Kevin McDonnell
Stony Brook University – CSE 215
27