Functions

1
Functions

• A function f from a set X to a set Y is a
  relation from X to Y such that x ? X is related
  to one and only one y ? Y
• X is called the domain Y is called the range.
  We say x is mapped into y.
• f (a, 3), (b, 3), (c, 5), (d, 1)

2
Function description

• Functions are described as
• Set of ordered pairs, example f as given before
• Using a formula such as f(x) expression,
  example f (x) 2x 2
• Illustration as in the example below

3
Properties of functions

• A function is called one to one or (injection) if
  different elements in the domain are related to
  different elements in the range.

4
Properties of functions

• A function is called onto or a surjection if each
  element of the range is related to at least one
  element in the domain.

5
Properties of functions

• A function is called bijection if it is one to
  one onto
• For each element in the domain there is one and
  only one element in the range
• Number of elements in the domain equals that in
  the range

6
Sequences

• A sequence is a function with domain is a set of
  consecutive integers.
• Example S(i) i2 for i 1
• The ith term of the sequence i2 is denoted by Si
  
• What is the second term of the sequence 22 4
• The fourth is 42 16

7
Finite infinite Sequences

• If the domain of the sequence is finite, then the
  elements can be listed as a tuple for example
• If the domain of the previous example S(i) is 1,
  2, 3,., 9, 10, then the sequence elements are
  1, 4, 9, 16, 25, 36, 49, 64, 81, 100
• The domain of S(i) is the integers, in this case
  the sequence elements are infinite 1, 4, 9, ,
  i2, .,

8
Summing Finite Sequence

• Example If n 3, then the sum 14
• Is there a formula to compute the sum of finite
  sequence elements? For the example S(i).
• How about the sequence 1 2 3 . n

9
Identity function

• For any domain X, the identity function IX?X is
  the unique function such that ?a?X I(a)a.
• Note that the identity function is both
  one-to-one and onto (bijective).

10
Identity Function Illustrations

• The identity function

Domain and range
11
Other Functions

• In discrete math, we will frequently use the
  following functions over real numbers
• ?x? (floor of x) is the largest (most positive)
  integer ? x.
• ?x? (ceiling of x) is the smallest (most
  negative) integer ? x.

12
Visualizing Floor Ceiling

• Real numbers fall to their floor or rise to
  their ceiling.
• Note that if x?Z,??x? ? ? ?x? ??x? ? ? ?x?
• Note that if x?Z, ?x? ?x? x.

3
.
?1.6?2
2
.
1.6
.
1
?1.6?1
0
.
??1.4? ?1
?1
.
?1.4
.
?2
??1.4? ?2
.
.
.
?3
?3
??3???3? ?3
13
Cartesian graphs of relations Functions

• Discovering properties of relations and functions
  from graphs
• Example

This is a relation but not a function vertical
line crosses the graph in two points
14
Plotting floor function

• Plot of graph of function f(x) ?x/3?

f(x)
Set of points (x, f(x))
2
x
?3
3
?2
15
Comparing functions

• A function f gt g for finite number of points but
  less than for others
• Example the identity function and x2

16
Growth rate function

• The time needed to execute a program depends on
  several factors among them the number of input
  values.
• If T is the running time then it is given as a
  function of number of input elements as T(n)
• For example if L is a list of elements it is
  needed to search L for some value. The running
  time depends on the size of L.

17
A linear time algorithm

• The running time T(n) is of the form of the line
  equation. T(n) an b
• Searching the list L for a given value
• Start with the first position
• Compare the key with the list element in the
  first position if found then, DONE and exit
• If not found try the next position and repeat the
  above step

18
A linear time algorithm

• What is the number of comparisons needed to find
  out that the key is not found in the list?
• What is the number of comparisons to find the key
  in the first position?
• What is the average time of finding the key using
  this algorithm?
• A program that implements the algorithm takes
  linear time.

19
One more linear algorithm

• Let L be a list of integer values. Is it possible
  to find the smallest element and then bring it to
  the first position?
• Start from the second position
• For each position i from 2 to n
• If the element in the ith position is less than
  the one in the first exchange it with the number
  in position 1
• Next i

Continue
20
Linear algorithm

• What is the number of comparisons that will be
  made to accomplish the task of producing the
  list with the smallest number in the first
  position? If each comparison needs a time then,
  the time needed will be (n-1)a
• If each exchange needs b time, then
• If no exchange takes place, then T(n)(n-1)a
• If the exchange takes place each time, then
• T(n) (n-1)a (n-1)b (n-1)(ab)
• So (n-1)a T(n) (n-1)(ab)

21
A sorting algorithm

• If the process in the previous example is
  repeated and the smallest integer in the
  remaining values is placed in the second
  position, then in the same way this process needs
  n-2 number of comparisons.
• Repeating the process for the third, fourth so
  on. Leads to SORTING the list in ascending order
  

22
T(n) for the sorting algorithm

• T(n) in the minimum needs
• (n-1)a (n-2)a (n-3)a 1a
• T(n) in the maximum needs
• (n-1)(ab) (n-2)(ab) 1(ab)

23
Comparison between the algorithms

• How the two algorithms are compared?
• If T1(n) an b (linear)
• T2(n) an2 bn c (quadratic), then
• how do they compare?
• Example T1(n) 100n 50 and T2(n) n2.

24
Order of running time

• A function f(n) is of order O(g(n) if there is a
  constant c and a number n0 such that for all n gt
  n0, f(n) c g(n)
• Running time functions are ordered using the O
  notation.
• Example T1 comes before T2 in the previous
  example.
• The order of the functions constant, log,
  linear, nlog(n), n2, n3, .2n, .