Functions, Sequences, and Sums

Functions, Sequences, and Sums

2.3 Functions Introduction • Function: task, subroutine, procedure, method, mapping, … • E.g. Find the grades of student x. intfindGrades(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 • We can use a formula or a computer program to define a function. Example: f(x) = x + 1 described as: intincreaseByOne(int x){ x = x + 1; return x; } 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.

2.3 Functions • 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 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 fmapsA to B.

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 • Example: Let f1and f2be functions from R to R such that f1(x) =x2 and f2 (x) = x – x2. What are the functions f1 + f2 and f1f2 ? Solution: (f1 + f2 )(x) = f1(x) + f2 (x) = x2 + (x – x2) = x (f1f2 ) (x) = f1(x) f2 (x) = x2(x – x2) = x3 –x4 DEFINITION 3 If f1and f2be 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) (f1f2) (x) = f1(x) f2 (x)

2.3 Functions One-to-One and Onto Functions • 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. 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)

2.3 Functions DEFINITION 7 A function f from A to B is called onto, or surjective, if and only if for every element bB there is an element aA 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 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 DEFINITION 8 The function f is a one-to-one correspondence or a bijection, if it is both one- to-one and onto.

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 • 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 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.

2.4 Sequences and Summations It is analogue of the linear function f(x) = dx+a • 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 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.

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 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 (inti=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 (inti=1; i<=4; i++){ sum2 = 0; for (intj=1; j<=3; j++){ sum2 = sum2 + i*j; } sum1 = sum1 + sum2; }

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

Functions, Sequences, and Sums