Func%ons (Math) Func%on Defini%ons. Func%ons (Computer Sciences) Func%on = mappings or transforma%ons Math examples:

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1 Func%ons (Math) Functions and Sequences (Rosen, Chapter.-.) TOPICS Functions Cartesian Products Sequences Geometric Progressions Func%on = mappings or transforma%ons Math examples: f(x) = x f(x) = x + f(x) = x f(x) = x 0//0 CS60 Fall Semester 0 Func%on Defini%ons A func%on f from sets A to B assigns exactly one element of B to each element of A. Example: the floor func%on Domain A 5 6 Codomain B Range: {,,,5} What s the difference between codomain and range? Range contains the codomain values that A maps to 0//0 CS60 Fall Semester 0 Func%ons (Computer Sciences) In Programming Func%on header defini%on example int floor(float real) { } Domain = R Codomain = Z 0//0 CS60 Fall Semester 0

2 Other Func%ons The identy func%on, ƒ ID, on A is the func%on where: ƒ ID (x) = x for all x in A. A = {a,b,c} and f(a) = a, f(b) = b, f(c) = c The successor function, ƒ succ (x) = x+, on Z, maps a number into the number following it. f() = f(-7) = -6 f(a) Does NOT map to b Only works on set Z The predecessor function, ƒ pred (x) = x-, on Z, maps a number into the number before it. f() = 0 f(-7) = -8 Other Func%ons ƒ NEG (x) = - x, also on R (or Z), maps a value into the nega%ve of itself. ƒ SQ (x) = x, maps a value, x, into its square, x. The ceiling func%on: ceil(.) =. 0//0 CS60 Fall Semester 0 5 0//0 CS60 Fall Semester 0 6 Func%ons What is NOT a func%on? What are ceiling and floor useful for? Data stored on disk are represented as a string of bytes. Each byte = 8 bits. How many bytes are required to encode 00 bits of data? Need smallest integer that is at least as large as 00/8 00/8 =.5 But we don t work with ½ a byte. So we need bytes 0//0 CS60 Fall Semester 0 7 Consider ƒ SQRT (x) from Z to R. This does not meet the given defini%on of a func%on, because ƒ SQRT (6) = ±. In other words, ƒ SQRT (x) assigns exactly one element of Z to two elements of R. No Way! Say it ain t so!! Note that the conven%on is that x is always the posi%ve value. ƒ SQRT (x) = ± x 0//0 CS60 Fall Semester 0 8

3 to Func%ons to Func%ons, cont. A func%on f is said to be one- to- one or injecve if and only if f(a) = f(b) implies that a = b for all a and b in the domain of f. Example: the square func%on from Z + to Z Is square from Z to Z an example? NO! Because ƒ SQ (- ) = = ƒ SQ (+)! Is floor an example? INCONCEIVABLE!! Is identy an example? Unique at last!! How dare they have the same codomain! 0//0 CS60 Fall Semester 0 9 0//0 CS60 Fall Semester 0 0 Increasing Func%ons A func%on f whose domain and co- domain are subsets of the set of real numbers is called increasing if f(x) <= f(y) and strictly increasing if f(x) < f(y), whenever x < y and x and y are in the domain of f. Is floor an example?.5 <.7 and floor(.5) = = floor(.7) and. <. and floor(.) = < = floor(.), Is square an example? So YES floor is an increasing function when mapping Z to Z or R to R, as square(-) = > yet square () where - < BUT it is NOT STRICTLY increasing. NO square is NOT an increasing function UNLESS. Domain is restricted to 0//0 CS60 Fall Semester 0 positive # s How is Increasing useful? Most programs run longer with larger or more complex inputs. Consider the maze: Larger maze may (in the worst case) take longer to get out. Maze with more walls may (in the worst case) take longer to get out. Consider looking up a telephone number in the paper directory 0//0 CS60 Fall Semester 0

4 Cartesian Products and Func%ons A func%on with mul%ple arguments maps a Cartesian product of inputs to a codomain. Example: Math.min maps Z x Z to Z int minval = Math.min(, 99 ); Find the minimum value between two integers Math.abs maps Q to Q+ int absval = Math.abs( - ); Find the absolute value of a number 0//0 CS60 Fall Semester 0 Quiz Check Is the following an increasing func%on? Z à Z f(x) = x + 5 Z à Z f(x) = x - 0//0 CS60 Fall Semester 0 Sequences An example sequence Your book s definition: A sequence is a function from a subset of the integers (usually {,, }) to a set S. In other words: it s an ordered set, where a function describes the mapping from position (,, ) to element a n = n,, What happens to a n when n gets large? a n gets small, but stays positive 0//0 CS60 Fall Semester 0 5 0//0 CS60 Fall Semester 0 6

5 Example Sequences: Sequence {a n } where a n = n What is this sequence? 0, -, -, -, What is the sum of the first 5 terms? 0 + (- ) + (- ) + (- ) + (- ) = - Important Sequences: Arithme%c Progressions: S = a, a + d, a + d,, a + nd, Example: write a loop that. Reads in a new string from the terminal. Compares it to every previous string, to see if it is new How many comparisons does this program do? 0//0 CS60 Fall Semester 0 7 0//0 CS60 Fall Semester 0 8 Nota%on: Σ Geometric Progressions Ajer N itera%ons, the number of comparisons is: (N- ) n This is wriken as: i i= 0 In English: the sum of i from i=0 to i=n- Ques%on: what is the sum from 0 to N-? Try with N=5, N=0. See if you can arrive at a general formula. 0//0 CS60 Fall Semester 0 9 Geometric Progression: S = a, ar, ar,, ar n, Example: how many nodes in a binary tree? Answer: n + New ques%on: what is this sum from to N? Try with n=5 0//0 CS60 Fall Semester 0 0 5