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

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

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; }

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

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}

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} ..

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

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

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.

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.

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

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 20-Nov-18 Computer Science Department

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

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.

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.

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))

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.

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

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.

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.

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

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 = -1 + 4n 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

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

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

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

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.14 b.11 c.15 d.23 d c

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:

2.4 Sequences and Summations What is the value of ? Solution: = 1 + 4 + 9 + 16 + 25 = 55 Expressed with a for loop: int sum = 0; for (int i =1; i <=5; i++) { sum = sum + i * i ; }

2.4 Sequences and Summations What is the value of the double summation ? Solution: = = = 6 + 12 + 18 + 24 = 60

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:

Exercises Write ( 1 + 4 + 9 + 16 + ... + 49 ) using sigma notation . Solution: 𝑘=1 7 𝑘 2 20-Nov-18 Computer Science Department

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