Functions, Sequences, and Sums

2.3 Functions Introduction Function: task, subroutine, procedure, method, mapping, … E.g. Find the grades of student x. int findGrades(string name){ //go to grades array, //find the name, and find the corresponding grades … return grades; } Adams A Chou B Goodfriend C Rodriguez D Stevens F

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 codomain 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

2.3 Functions 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}

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. Then, 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}

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 (f1 + f2 )(x) = f1(x) + f2 (x) 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. a 1 b 2 c 3 d 4 5 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? a 1 b 2 c 3 d 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. a. One-to-one, b. Onto, c. One-to-one, d. neither d. Not a Not onto not one-to-one and onto function a 1 a a 1 a 1 1 b 2 b 1 b 2 b 2 a 2 c 3 c 2 c 3 c 3 b 3 4 d 3 d 4 d 4 c 4

2.4 Sequences and Summations 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

2.4 Sequences and 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;

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