Lecture 4 Sequences CSCI 1900 Mathematics for Computer

Lecture 4 Sequences CSCI – 1900 Mathematics for Computer Science Fall 2014 Bill Pine

Lecture Introduction • Reading – Rosen - Section 2. 4 • Sequences – Definition – Properties • Recursion • Characteristic Function CSCI 1900 Lecture 4 - 2

Sequence • A sequence is an ordered list of objects • It can be finite – 1, 2, 3, 2, 1, 0, 1, 2, 3, 1, 2 Or infinite – 3, 1, 4, 1, 5, 9, 2, 6, 5, … • In contrast to sets, with sequences – Duplicates are significant – Order is significant CSCI 1900 Lecture 4 - 3

Sequence Examples • 1, 2, 3, 2, 2, 3, 1 is a sequence, but not a set • The sequences 1, 2, 3, 2, 2, 3, 1 and 2, 2, 1, 3 are made from elements of the set {1, 2, 3} • The sequences 1, 2, 3, 2, 2, 3, 1 and 2, 1, 3, 2, 2, 3, 1 only switch two elements, but that is sufficient to make them unequal CSCI 1900 Lecture 4 - 4

Alphabetic Sequences • Finite list of characters – D, i, s, c, r, e, t, e – d, a, b, d, c, a • Infinite lists of characters – a, b, a, b… – This, the, song, that, never, ends, It, goes, on, and, on, my, friends, Someone, started, singing, it, not, knowing, what, it, was, and, they’ll, continue, singing, it, forever, just, because… • String : a sequence of letters or symbols written without commas CSCI 1900 Lecture 4 - 5

Linear Array • Principles of sequences can be applied to computers, specifically, arrays • There are some differences though – Sequence • Well-defined • Modification of any element or its order results in a new sequence – Array • May not have all elements initialized • Modification of array by software may occur • Even if the array has variable length, we’re ultimately limited to finite length CSCI 1900 Lecture 4 - 6

Set Corresponding to a Sequence • Set corresponding to a sequence - the set of all distinct elements of the sequence – {a, b} is the set corresponding to sequence abab… – {1, 2, 3} is the set corresponding to the sequences • 1, 2, 3, 3, 2, 1 • 1, 2, 3, 3, 2, 1, 1, 2, 3, 3, 2, 1…. CSCI 1900 Lecture 4 - 7

Defining Infinite Sequences • Explicit Formula, – The value for the nth item – an – is determined by a formula based solely on n • Recursive Formula – The value for the nth item is determined by a formula based on n and the previous value(s) – an , an-1 , … – in the sequence CSCI 1900 Lecture 4 - 8

Explicit Sequences • Easy to calculate any specific element • By convention a sequence starts with n = 1 • an = 2 * n – 2, 4, 6, 8, 10, 12, 14, 16, … – a 3 = 2 * 3 = 6 – a 5 = ? ? ? • an = 3 * n + 6 – 9, 12, 15, 18, 21, 24, … – a 5 = 3 * 5 + 6 = 21 – a 100 = ? ? ? • Defines the values a variable is assigned in a program loop when adding/subtracting/multiplying/dividing by a constant CSCI 1900 Lecture 4 - 9

Explicit Sequences and a Program for n = 1 thru 100 by 1 y = n * 5 display “y sub” n next n “is equal to” y yn = 5 *n in this sequence CSCI 1900 Lecture 4 - 10

Recursive Sequences • Used when the nth element depends on some calculation on the prior element(s) • You must specify the values of the first term(s) • Example: a 1 = 1 a 2 = 2 an = an-1 + an-2 • Defines the values a variable is assigned in a program loop when the current value depends upon its previous value(s) CSCI 1900 Lecture 4 - 11

Recursive Sequences and a Program y. Minus 2 = 1 print “y sub 1 is equal to” y. Minus 2 y. Minus 1 = 2 print “y sub 2 is equal to” y. Minus 1; for n = 3 thru 100 by 1 y = y. Minus 1 + y. Minus 2 print “y sub” n “ is equal to” y y. Minus 2 = y. Minus 1 = y next n yn = yn-1+yn-2 in this sequence CSCI 1900 Lecture 4 - 12

Characteristic Functions • Characteristic function – function defining membership in a set, over the universal set • f. A(x) = 1 if x A 0 if x A • Example A = {1, 4, 6} Where U = Z+ – f. A(1) = 1, f. A(2) = 0, f. A(4) = 1, f. A(5) = 0, f. A(6) = 1, f. A(7) = 0, … – f. A(0) = ? ? ? CSCI 1900 Lecture 4 - 13

Representing Sets with a Computer • Recall: sets have no order and no duplicated elements • The need to assign each element in a set to a memory location – Gives the set order in a computer • We can use the characteristic function to define a set with a computer program CSCI 1900 Lecture 4 - 14

Characteristic Function in Programming • To see a simplistic implementation, consider the following example with A ={1, 4, 6} function is. In. A( x ) if( (x == 1) OR (x == 4) OR (x == 6) ) return ( 1 ) else return ( 0 ) • Might be used to validate user input CSCI 1900 Lecture 4 - 15

Properties of Characteristic Functions Characteristic functions of subsets satisfy the following properties (proofs are on page 15 of textbook) – f. A B(x) = f. A(x)f. B(x) – f. A B(x) = f. A(x) + f. B(x) – 2 f. A(x)f. B(x) CSCI 1900 Lecture 4 - 16

Proving Characteristic Function Properties • A way to prove these is by enumeration of cases • Example: Prove f. A B = f. Af. B a A c d B b f. A(a) = 0, f. B(a) = 0, f. A(a) f. B(a) = 0 0 = f. A B(a) f. A(b) = 0, f. B(b) = 1, f. A(b) f. B(b) = 0 1 = 0 = f. A B(b) f. A(c) = 1, f. B(c) = 0, f. A(c) f. B(c) = 1 0 = f. A B(c) f. A(d) = 1, f. B(d) = 1, f. A(d) f. B(d) = 1 1 = f. A B(d) CSCI 1900 Lecture 4 - 17

More Properties of Sequences • Any set with n elements can be arranged as a sequence of length n, but not vice versa – Sets have no order and duplicates don’t matter – Each subset can be identified with its characteristic function – A sequence of 1’s and 0’s • Characteristic function for universal set, U, is f. U(x) = 1 CSCI 1900 Lecture 4 - 18

Countable and Uncountable • A set is countable if it corresponds to some sequence – Members of set can be arranged in a list – Elements have position • All finite sets are countable • Some, but not all infinite sets are countable, otherwise are said to be uncountable – An example of an uncountable set is set of real numbers – E. g. , what comes after 1. 23534 ? CSCI 1900 Lecture 4 - 19

Countable Sets • A finite set S is always countable – It can correspond the sequence • an = n for all n N where n |S| • Countable { x | x = 5 * n and n Z+ } – Corresponds to the sequence an = n for all Set 5 10 15 20 25 30 Seq 1 2 3 4 5 6 CSCI 1900 Lecture 4 - 20

Strings • Sequences can be made up of characters • Example: W, a, k, e, , u, p • Removing the commas and you get a string “Wake up” • Strings can illustrate the difference between sequences and sets more clearly – a, b, a, … is a sequence, i. e. , “abababa…” is a string – The corresponding set is {a, b} CSCI 1900 Lecture 4 - 21

Key Concepts Summary • Sequences – Integers – Strings • Recursion • Characteristic Function CSCI 1900 Lecture 4 - 22