1. CS100: Discrete structures
Computer Science Department
Lecture 2: Functions, Sequences, and Sums
Ch2.3, Ch2.4

2. 2.3 Function introduction :
Function: task, subroutine, procedure, method, mapping, …
E.g. Find the grades of student x.
char findGrades(string name){
//go to grades array,
//find the name, and find the corresponding grades …
return grades; }
20-Nov-18
Computer Science Department

3. 2.3 Functions
DEFINITION 1
Let A and B to be nonempty sets. a function f from A to B is an assignment
of exactly one element of B to each element of A. We write f(a) = b if b is the
unique element of B assigned by the function f to the element a of A. If f is a
function from A to B, we write f: A → B.
We can use a formula or a computer program to define a function.
Example:
f(x) = x + 1
described as:
int increaseByOne(int x){
x = x + 1;
return x; }

4. 2.3 Functions
DEFINITION 2
If f is a function from A to B, we say that A is the domain of f and B is the
Co-domain of f. If f(a) = b, we say that b is the image of a and a is a preimage
of b. The range of f is the set of all images of elements of A. Also, if f is a
function from A to B, we say that f maps A to B.
For each function, we specify its domain, codomain and the mapping of elements of the domain to elements in the codomain
Two functions are said to be equal if they have the same domain, codomain, and the map elements of their common domain to the same elements of their common codomain
A function is differ by changing its domain, codomain or the mapping of elements

5. Exercise ..
What are the domain, codomain, and range of the function that assigns grades to students described in the slide 2?
Solution:
domain: {Adams, Chou, Goodfriend, Rodriguez, Stevens}
codomain: {A, B, C, D, F}
range: {A, B, C, F}

6. 2.3 Functions - Example Solution:
Let f be the function that assigns the last two bits of a bit string of length 2 or greater to that string. For example, f (11010) = 10.
What is the domain, codomain and range of the function ?
Solution:
the domain of f is the set of all bit strings of length 2 or greater ..
and both the codomain and range are the set {00,01,10,11} ..

7. 2.3 Functions - Example
What is the domain and codomain of the function :
int floor(real float){…}?
Solution:
domain: the set of real numbers
codomain: the set of integer numbers

8. 2.3 Functions What are the functions f1 + f2 and f1 f2 ?
DEFINITION 3
If f1 and f2 be functions from A to R. Then f1 + f2 and f1 f2 are also functions
from A to R defined by
(f1 + f2 )(x) = f1(x) + f2 (x)
(f1 f2 ) (x) = f1(x) f2 (x)
Example: Let f1 and f2 be functions from R to R such that f1 (x) =x2 and f2 (x) = x – x2.
What are the functions f1 + f2 and f1 f2 ?
Solution:
(f1 + f2 )(x) = f1(x) + f2 (x) = x2 + (x – x2) = x
(f1 f2 ) (x) = f1(x) f2 (x) = x2(x – x2) = x3 – x4

9. 2.3 Functions One-to-One and Onto Functions
DEFINITION 5
A function f is said to be one-to-one, or injective, if and only if
f(a) = f(b) implies that a = b for all a and b in the domain of f. A function is said to be an injection if it is one-to-one. (every element in the range is a unique image for element of A – all image have at most one arrow or none)
a , b(a ≠ b → f(a) ≠ f(b)) (If it’s a different element, it should map to a different value.)
Example: Determine whether the function f from {a,b,c,d} to {1,2,3,4,5} with f(a) = 4, f(b) = 5, f(c) = 1 and f(d) = 3 is one-to-one.
Solution: Yes.

10. 2.3 Functions
DEFINITION 7
A function f from A to B is called onto, or surjective, if and only if for every
element b B there is an element a A with f(a) = b. a function f is called a
surjection if it is onto. Co-domain = range
Example: Let f be the function from {a,b,c,d} to {1,2,3} defined by
f(a) = 3, f(b) = 2, f(c) = 1, and f(d) = 3. Is f an onto function?
Solution: Yes.
Example: Is the function f (x) = x2 from the set of integers to the set of integers onto?
Solution: No. There is no integer x with x2 = -1, for instance.

11. 2.3 Functions
DEFINITION 8
The function f is a one-to-one correspondence or a bijection, if it is both one-
to-one and onto.
One-to-One
Not onto
Onto
Not One-to-One
One-to-One
And onto
bijection
Not One-to-One
Not onto
(Neither)
Not Function

12. Exercise .. a) f(x) = 2x + 1 b) f(x) = x2 + 1 c) f(x) = x3
Determine whether each of these functions is a bijection from R to R.
a) f(x) = 2x + 1
b) f(x) = x2 + 1
c) f(x) = x3
d) f(x) = (x2 + I )/(x2 + 2)
Yes No
Yes No
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Computer Science Department

13. Exercise .. Why f is not a function from R to R ? f(x)=
f(0) is not defined
f(x) is not defined for x<0
f(x) is not function because there are two values assigned to each x.
20-Nov-18
Computer Science Department

14. 2.3 Functions
DEFINITION 9
Let f be a one-to-one correspondence from the set A to the set B. The inverse function of f is the function that assigns to an element b belonging to B the unique element a in A such that f(a) = b.
The inverse function of f is denoted by f-1. Hence, f-1(b)=a when f(a) = b.
A one-to-one correspondence is called invertible because we can define an inverse of this function.
A function is not invertible if it is not a one-to-one correspondence because the inverse of such function does not exist.

15. 2.3 Functions
Example: Let f be the function from {a, b, c} to {1, 2, 3} such that f(a) =2, f(b) = 3 and f(c) = 1. Is f invertible? And if it is, what is its inverse?
Solution: Yes, it is invertible, because it is a one-to-one correspondence.
f-1(1) = c, f-1(2) = a, f-1(3) = b.
Exercise: Let f : Z Z be such that f(x) = x+1. Is f invertible? And if it is, what is its inverse?
Solution: Yes, it is invertible, because it is a one-to-one correspondence.
f-1(y) = y-1.
Exercise: Let f be a function from R to R with f(x) = x2. Is f invertible?
Solution: f is not one-to-one because f(-2) = f(2) = 4. So, it is not one-to-one correspondence and hence it is not invertible.

16. 2.3 Functions
DEFINITION 10
Let g be a function from the set A to the set B , and let f be a function from the set B to the set C . The composition of the functions f and g, denoted by f o g, is defined by: (f o g)(a) =  f(g(a))

17. 2.3 Functions
f o g is the function that assigns to the element a of A the element assigned
by f to g(a). That is, to find (f o g )(a) we:
Apply the function g to a to obtain g(a)
Then we apply the function f to the result g(a) to obtain (f o g)(a) =  f(g(a))
NOTE: The composition f o g cannot be defined unless the range of g is a subset of the domain of f.

18. 2.3 Functions
Example:
Let g be the function from the set {a , b, c} to itself such that
g(a) = b, g(b) = c, and g(c) = a
Let f be the function from the set {a , b , c} to the set { l , 2 , 3 } such that
f(a) = 3 , f(b) = 2 , and f(c) = 1
What is the composition of f and g, and what is the composition of g and f?
Solution:
The composition f o g is defined by:
(f o g)(a) = f(g(a)) = f(b) =2
(f o g)(b) = f(g(b)) = f(c) =1
(f o g)(c) = f(g(c)) = f(a) = 3
g o f is not defined, because the range of f is not a subset of the domain of g

19. 2.3 Functions
Example:
Let f and g be the functions from the set of integers to the set of integers defined by f(x) = 2x + 3 and g(x) = 3x + 2.
What is the composition of f and g? What is the composition of g and f?
Solution:
Both the compositions f o g and g o f are defined
(f o g)(x) = f(g(x)) = f(3x + 2) = 2(3x + 2) + 3 = 6x + 7
(g 0 f)(x) = g(f(x)) = g(2x + 3) = 3(2x + 3) + 2 = 6x + 11.

20. 2.4 Sequences and Summations Sequences
A sequence is a discrete structure used to represent an ordered list
Example: 1,2,3,5,8
1,3,9,27,81,…,30,…
We use the notation {an} to denote the sequence.
Example: Consider the sequence {an}, where an = 1/n.
The list of the terms of this sequence, beginning with a1, namely
a1, a2, a3, a4, …, starts with 1, 1/2, 1/3, 1/4, …
DEFINITION 1
a sequence is a function from a subset of the set of integers (usually either
the set {0,1,2,…} or the set {1,2,3,…}) to a set S. We use the notation an to
denote the image of the integer n. We call an a term of the sequence.

21. 2.4 Sequences and Summations
It is analogue of the exponential function f(x) = arx
DEFINITION 2
a geometric progression is a sequence of the form
a, ar, ar2, …, arn, …
where the initial term a and the common ratio r are real numbers.
Example: The following sequence are geometric progressions.
{bn} with bn = (-1)n starts with 1, -1, 1, -1, 1, …
initial term: 1, common ratio: -1
{cn} with cn = 2*5n starts with 2, 10, 50, 250, 1250, …
initial term: 2, common ratio: 5
{dn} with dn = 6 *(1/3)n starts with 6,2, 2/3, 2/9, 2/27, …
initial term: 6, common ratio: 1/3

22. 2.4 Sequences and Summations
It is analogue of the linear function
f(x) = dx+a
DEFINITION 3
A arithmetic progression is a sequence of the form
a, a + d, a + 2d, …, a + nd, …
where the initial term a and the common difference d are real numbers.
Example: The following sequence are arithmetic progressions.
{sn} with sn = n starts with -1, 3, 7, 11,…
initial term: -1, common difference: 4
{tn} with tn = 7 – 3n starts with 7, 4, 1, -2, …
initial term: 7, common difference: -3

23. 2.4 Sequences and Summations
Example :
Find formulae for the sequences with the following first five terms
(a). 1, 1/2, 1/4, 1/8, 1/16
Solution: an = 1/2n
(b). 1, 3, 5, 7, 9
Solution: an = (2n )+ 1
(c). 1, -1, 1, -1, 1
Solution: an = (-1)n

24. Exercises Find the formula of this sequence and find A6 , A8 :
It is arithmetic sequence ,
initial term a= 20 ,
common difference d= 4 ,
an=10+4n
A6 = 34 , A8 = 42

25. Exercises A = 4, 8, 16, 32 It is geometric sequence ,
initial term = 4 ,
common ratio = 2 ,
an=4*2n
A6 = 256, A8 = 1024

26. Exercises
in a geometric sequence a0 was =3 and r =1/2 then a3 is equal to ‎
a. 3/16 ‎ b. 3/4‎
c. 3/2‎ d. 3/8‎ ‎
In an arithmetic sequence a0 was =7 and a3 was 19 the value of a2 is=
a b.11
c d.23 d c

27. 2.4 Sequences and Summations Summations
The sum of the terms from the sequence am + am+1, …, an can be expressed as
, Or
Where m is the lower limit, n is the upper limit, and j is the index of the summation
Example:
Express the sum of the first 100 terms of the sequence {an}, where
an = 1/n for n = 1,2,3, ….
Solution:

28. 2.4 Sequences and Summations
What is the value of ?
Solution:
= = 55
Expressed with a for loop:
int sum = 0;
for (int i =1; i <=5; i++) {
sum = sum + i * i ; }

29. 2.4 Sequences and Summations
What is the value of the double summation ?
Solution: =
= = = 60

30. Sequences and Summations
Expressed with two for loops:
int sum1 = 0;
int sum2 = 0;
for (int i =1; i <=4; i ++){
sum2 = 0;
for (int j=1; j<=3; j++){
sum2 = sum2 + i *j; }
sum1 = sum1 + sum2;
Solution:

31. Exercises Write ( 1 + 4 + 9 + 16 + ... + 49 ) using sigma notation .
Solution:
𝑘=1 7 𝑘 2
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Computer Science Department

32. ANY QUESTIONS??
Refer to chapter 2 of the book for further reading