Section 4.2
Functions
Announcement
• Exam #3 has been rescheduled
– Monday, April 4th
– Chapters 3 and 4
– Practice sheet for Chapter 3 now posted.
Activity 1
Let f: X → Y as shown by :
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What is the domain of f?
What is the target of f?
What is the range of f?
What is f(a)? f(b)? f(c)?
Review
• What is a function?
Definition of a function
• A function f from a set X to a set Y, denote
f: X → Y, is a relation from X, the domain
to Y, the target (or 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
Consider the following
• Let
N = {Adams, Epp, Stevens, Rosen, Smedley}
L = {A, B, C, D, F}
g : N → L such that the mapping indicates the
student’s grade in the class.
Is g a function?
Non-functions
• What kind of situations would need to occur
for this to not be a valid function.
We sometimes write functions as
a set of ordered pairs.
g = {(Adams, A), (Epp, C), (Stevens, B),
(Rosen, A), (Smedley, F)}
Activity #2
• Represent f as a set of ordered pairs.
Activity 3
Consider:
L = {1,2,3,4,5}
R = {A,B,C,D,F}
Which of the following are functions from L to R?
1. {(1,A),(2,B),(3,C),(4,D),(5,F)}
2. {(2,A),(2,B),(3,C),(4,D),(5,F)}
3. {(1,A),(2,A),(3,B),(4,B),(5,B)}
4. {(1,5),(2,4),(3,3),(4,2),(5,1)}
5. {(2,A),(3,B),(4,B),(5,B)}
Activity 4
• Find all of the functions from X = {a,b} to
Y={u,v}
• Find all of the functions from X = {a,b} to
Y={u}
• How many functions would exist from X =
{a,b,c,d} to Y={u,v,x}
• How does this relate back to Cartesian
products of sets?
Common Functions in CS
• Floor function
floor: R → Z
floor(x) = the largest integer y such that y≤x
Common Functions in CS
• Ceiling function
ceiling: R → Z
ceiling(x) = the smallest integer y such that
x≤y
Common Functions in CS
• Encoding and Decoding functions
E: bitString → bitString
D: bitString → bitString
Common Functions in CS
• Encoding and Decoding functions
E(s) = the string obtained from s by replacing
each bit of s by the same bit written three
times
D(t) = the string obtained from t by replacing
each consecutive triple of three bits by a
single copy of that bit
Common Functions in CS
• Encoding and Decoding functions
E(s) = the string obtained from s by replacing
each bit of s by the same bit written three
times
D(t) = the string obtained from t by replacing
each consecutive triple of three bits by
whichever bit is most frequent.
Common Functions in CS
• The Hamming Distance
H: bitString,bitString → Z
H(s,t) = the count of the number of positions
in which s,t have different values