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Chapter 7 Functions In Layman’s terms A function f from a set X to another set Y is a recipe that tells you how to convert an input number x to the output number y. Usually such a recipe is expressed in a formula, but we also have: Function defined by arrow diagrams f X 1 a b c 2 3 4 Note: this method works only for finite sets. Y Arrow diagram of a function with animation. 2 a b 3 5 8 c 13 d 19 e 26 The following diagram does not represent a function, do you know why? 2 a b 3 5 8 c 13 d 19 e 26 Function as a set a A 2 f b c d e 3 5 8 13 B 19 26 The above function is defined as a set of ordered pairs: f = {(a, 5), (b, 26), (c, 2), (d,8), (e, 13)} and this set is a subset of A×B 7.1 Functions Defined on General Sets Definition: Given any two sets X and Y, a function f from X to Y is a subset of the Cartesian product X ×Y such that (a) for every a X, there is one y Y such that (a, y) f (b) if (a, y) f and (a, z) f , then y = z (in other words, there is one and only one output for each input) The set X is called the domain of f , Y is called the codomain, and range of f = {b Y : b = f (a) for some a X} Given any b Y, inverse image of b = f -1(b) = {a X : f (a) = b} and this set can be empty. Functions defined on the set of bit strings Let Σ = {0, 1} and Σ* be the set of all finite strings over Σ. (1) Define g: Σ* → Σ* by g(s) = the reverse of s from right to left. eg. g(10111) = 11101 (2) Define f: Σ* × Σ* → Σ* by f(s, t) = the concatenation of s followed by t. eg. f (000, 1101) = 0001101 Check digit functions The last digit of each UPC or ISBN is always created to catch errors. In other words, there is a formula to compute the last digit using all but the last digit as input. If this last digit does not match the formula, we know that the number must be wrong. Example: 1. In the UPC 0-53600-10054-0, the last digit 0 is chosen such that 0×3 + 5×1 + 3×3 + 6×1 + 0×3 + 0×1 + 1×3 + 0×1 + 0×3 + 5×1 + 4×3 + 0×1 is divisible by 10. In dot product notation, this can be written as [(0,5,3,6,0,0,1,0,0,5,4,0) (3,1,3,1,3,1,3,1,3,1,3,1)] MOD 10 = 0 2. Each ISBN is a 10 digit number such as 0-88385-720-0. The last digit a10 is chosen such that [(0,8,8,3,8,5,7,2,0,a10) (10,9,8,7,6,5,4,3,2,1)]MOD 11 = 0 In this case, a10 = 0 will work. And if a10 = 10, we use the letter X instead. In both examples, we see that there is always at most one output from each input, these procedures of generating check digits are functions even though they cannot be easily described by formulas. Coding functions A typical coding function usually assigns a unique numeric code (often a binary code) to each input. The three common reasons for assigning codes are (1) secrecy (or privacy), (2) data compression, (3) error correction. Examples of Error correcting Codes. Hamming [7,4,3] binary linear code. (1950) This is the 1st error detecting code (cannot correct error yet). Each input is a binary sting of length 4, and the function will add 3 check digits at the end and make the output a string of length 7. input 0000 0001 0010 0100 1000 output 0000000 0001011 0010111 0100101 1000110 input output 1100 1100011 1010 1010001 …. etc. It can detect 1 error out of the 7 bits, but cannot correct it. Functions defined by formulas Example: f(x) = 3x2 + 5 for every x in . g(x) = exsinx for every x in . 7 These functions can also be (partially) represented by graphs instead of arrow diagrams. 6 5 4 3 2 1 -5 -4 -3 x -2 -1 00 -1 1 2 3 Examples In mathematical analysis, many functions are defined by descriptions rather than by formulas. All examples below are functions from to . 1. This function is discontinuous everywhere. 0 if x is rational f ( x) 1 if x is irrational 1 Examples 2. This function is continuous only at 0. 0 if x is rational f ( x) x if x is irrational But it is not differentiable anywhere. 3. This function is differentiable at 0 but is discontinuous elsewhere. 0 if x is rational f ( x) 2 x if x is irrational 4. This function is continuous only at the irrationals. f x 1 q if x is rat ionaland x 0 if x is irrational p in reduced form q Remark: There is no function which is continuous only at the rational numbers. In 1872, Weierstrass published a paper that states that the following function is continuous but nowhere differentiable. f ( x) (0.5) n cos(13n x) n 0 (remark: He actually published a more general result.) Space Filling Curve In 1890, Giuseppe Peano discovered a continuous function from [0, 1] onto the unit square [0,1] ×[0,1]. However, this function cannot be one-to-one. The following function is due to David Hilbert, only the 1st 6 iterations are shown, the final function is the limit of these iterations. Equality of functions: Suppose that both f and g are functions from X to Y, then we say that f is equal to g, written f = g if and only if for every a X, f (a) = g(a) Example: f (x) = (x2 + x + 1) mod 3, g(x) = (x + 2)2 mod 3 They are equal on the domain of integers. Exercise Let f be a function from to such that 1 f ( x) 2 f ( )x 1 x whenever x 1. Find the value of f ( 12 ) Reed-Solomon Codes (1960) The commercially used version for CD’s, DVD’s, cellphones etc. is the [255,223,33]-code, in which (i) Every codeword is a 255-byte string, hence 2040 bits. (ii) In each codeword, 223 bytes are from the original message (others are check digits) (iii) It can correct up to 16 incorrect bytes (i.e. 16 bits in 16 different bytes in the worst case, and 16×8 bits in a row in the best case.) Finite-State Automata A finite state automaton is a machine that can make a few decisions, but it is much weaker than a computer because it does not have an expandable memory and it can only run one predetermined program. Examples: Vending machines, Definition: A finite-state automaton consists of five objects 1. a set I called the input alphabet, of input symbol; 2. a set S of states the automaton can be in; 3. a designate state s0 , called the initial state; 4. a designate set of states called the set of accepting states 5. a next-state function N : S × I S Finite-State Automaton Simple Vending Machine – accepts only 25¢ or 50¢ - gives a bottle of soda for $1 - does not return changes 25¢ desposited 0¢ desposited 50¢ 25¢ 25¢ 50¢ desposited 75¢ desposited 50¢ 50¢ $1 or more desposited 50¢ The Next-State Table Original State Input Quarter Half-Dollar 0¢ deposited 25¢ deposited 50¢ deposited 25¢ deposited 50¢ deposited 75¢ deposited 50¢ deposited 75¢ deposited Accept 75¢ deposited Accept Accept 25¢ deposited 50¢ deposited Accept Next State in blue Examples: 1. Construct a finite state automaton that accepts exactly the set of strings of 0’s and 1’s that start with the pattern 110. 2. Construct a finite state automaton that accepts exactly the set of strings of 0’s and 1’s for which the number of 1’s is divisible by 3. 3. Construct a finite state automaton that accepts exactly the set of strings of 0’s and 1’s that do not contain the pattern 1011. Question For any given set A of strings of 0’s and 1’s, can we build a finite-state automaton that accepts exactly the strings in the set A? In particular, can we build one machine that accepts exactly those strings where the number of 0’s is equal to the number of 1’s? And can we build one machine that accepts exactly those strings that are palindromes? (eg. 0110110) 7.2 1-to-1, Onto, and Inverse functions Definition: Let F be a function from a set X to a set Y. F is one-to-one (or injective) if, for all elements x1 and x2 in X F ( x1) F ( x2 ) x1 x2 Or equivalently, x1 x2 F ( x1) F ( x2 ) Different ways to check that a function is one-to-one Let f be a function from a connected interval in to . Then f is one-to-one if either of the following is true. 1.The graph of f passes the horizontal line test. 2. f is strictly increasing or strictly decreasing. 3. If f is differentiable, then f is one-to-one if f ’(x) > 0 for all x or if f ’(x) < 0 for all x in the domain of f. If f is defined on other domains, we have to use the definition to prove “one-to-one”ness, or we can construct an inverse for f. This is a one-to-one function. 2 a b 3 5 8 c d 13 19 e 26 This is not a one-to-one function. 2 a b 3 5 8 c d 13 19 e 26 We can call this a many-to-one function. Examples 1. The identity function id: defined by id (x) = x is one-to-one 2. Any linear function f(x) = ax + b with a ≠ 0 is one-to-one from to . Examples 3. Show that the function x f ( x) x5 is one-to-one throughout its domain. 4. Let S be the set of all finite strings of a’s, b’s and c’s. Define C: S S by C(s) = a*s for all sS C is called concatenation, i.e. C(bbc) = abbc . Show that C is one-to-one. Sets of Sequences Notations a. 2 is the set of all functions from Z+ to 2 (which is the set {0,1}) This is identified with the set of all infinite sequences of 0’s and 1’s. such as 0,1,0,1,0,1,0,1, … or 0,1,0,1,1,0,1,1,1,0, … b. 10 is the set of all functions from Z+ to 10 (which is the set {0,1, 2, 3, 4, 5, 6, 7, 8, 9}) Similarly this is identified with the set of all infinite sequences of digits; such as 3,2,4,8,0,5,9,1, … or 4,2,4,2,4,2,4,2,4,2, … More examples a. The embedding function I: 2 10 defined by I(s) = s for every s2 is one-to-one. b. We now try to construct a function G: 10 2 which is one-to-one. Properties of one-to-one functions. 1. If f(x): is one-to-one, and a, b, c are constants with a ≠ 0, b ≠ 0, then a·f(bx) + c is also one-to-one. 2. The composition of two or more one-to-one functions is also one-to-one. example: e sin(x) is one-to-one on the domain (-π/2, π/2). Onto functions Definition: Let F be a function from a set X to a set Y. F is onto (or surjective) if, range of F = Y Or equivalently, y Y x X ( F ( x) y) This function is not onto. 2 a b 3 5 8 c d 13 19 e 26 This function is onto (even though not one-to-one). 3 a b 8 c d e 19 Unfortunately there is no standard method to check whether a function is onto or not. Different functions may require different techniques. Whether a given function is onto or not onto depends on its co-domain as well. If we can reduce the co-domain, we can make a function onto. Examples The function f: defined by f (x) = 4x + 1 is onto, but the function f: Z+ Z+ defined by f (n) = 4n + 1 is not onto. More Examples c. The polynomial p(x) = x3 – 4x + 2 is onto but not one-to-one. d. There is an onto function H: 10 2 defined by 0 if f (n) is even H ( f )(n) for every f 10 1 if f (n) is odd but this function is not one-to-one. e. Challenge: Can you construct an onto function from 2 to 10 ? Theorem For any set S (finite or infinite), there is no onto function from S to (S). Proof: It is only necessary to consider the case where S is infinite, because if S is finite, we know that (S) has more elements. Assume to the contrary that there is a onto function f : S → (S) We then consider the subset A = { xS : xf(x) } Inverse functions Definition: Let f be a function from a set X to a set Y. If f is both one-to-one and onto, then we say that f is a bijection or a one-to-one correspondence between X and Y. Theorem: Suppose that f : X Y is a bijection, then there is a function f -1 : Y X defined by f -1 (y) = the unique element x in X such that f (x) = y Definition: The function f -1 defined above is called the inverse of f. Inverse functions Theorem: If X and Y are sets and f : X Y is one-to-one and onto, then f -1 : Y X is also one-to-one and onto. The following theorem provides a very convenient way to prove that a function is one-to-one and onto. Theorem: If X and Y are sets and f : X Y is a function. Suppose further that there is a function g: Y X such that g ◦ f = IX : X X then f is one-to-one and onto. Special cases Suppose that both X and Y are finite sets and f : X → Y is a function, (1) if n(X) > n(Y), then f cannot be one-to-one, (2) if n(X) < n(Y), then f cannot be onto, (3) if n(X) = n(Y), and f is one-to-one, then f is also onto, (4) if n(X) = n(Y), and f is onto, then f is also one-to-one. Example Let f : R → R be a continuous but non-constant function that preserves addition and multiplication, i.e. f (x + y) = f (x) + f (y) and f (x · y) = f (x) f (y) Prove that f is one-to-one. Solution: We need to divide the proof into several steps. (1) prove that f (0) = 0 (2) prove that f (-x) = - f (x) (3) prove that f (x) = 0 implies that x = 0. (4) prove that f (x) = f (y) implies that x = y. In fact, we can prove (in exercise) that this f is actually the identity function, hence it is also onto. 7.4 Cardinality with Applications Definition: A set S is called finite if there is no bijection between S and a proper subset of S. A set is called infinite if it is not finite. (In other words, a set S is infinite if we can construct a bijection from S to a proper subset of S. Example: The set of natural numbers N is infinite because we can construct the bijective function f(n) = n + 1 from N to a proper subset {1, 2, 3, 4, ··· } of N itself. Theorem: For any function f from a finite set X to a finite set Y, if n(X) > n(Y), then f cannot be one-to-one. This theorem is called the Pigeon hole Principle or the Dirichlet box principle. We shall study this principle in section 9.4 7.4 Cardinality and Applications Definition: Two sets A and B are said to have the same cardinality i.e. card(A) = card(B) if and only if there is a bijection between them. Using the terminology of cardinality, we can redefine infinite sets. Definition: A set S is said to be infinite if S has at least one proper subset W such that card(W) = card(S) Examples card({0,1,2,3,4,5, ··· }) = card ({0,2,4,6,8, ···}) = card () Exercises 1. Show that the interval (0, 1) has the same cardinality as the longer interval (5, 8). 2. Show that the interval (0, 1) has the same cardinality as the infinite interval (0, ∞). 3. Show that the interval (0, 1) has the same cardinality as the infinite interval (-∞, ∞). 4. Show that the closed interval [0, 1] has the same cardinality as the open interval (0, 1). 4. Show that the closed interval [0, 1] has the same cardinality as the open interval (0, 1). Solution: Consider the function 12 1 f ( x) n 2 x if x 0 if x 1n for a postive integer n. otherwise It is not difficult to check that f is one-to-one and it maps [0, 1] onto (0, 1) Theorem There is a one-to-one correspondence between the closed interval [0,1] and the closed square [0,1]×[0,1]. Proof: Let f : [0,1] → [0,1]×[0,1] be defined by f (0.a1a2a3a4a5a6…) = (0.a1a3a5… , 0.a2a4a6…) then f is bijective because it has an obvious inverse. (remark: we use infinite decimal expansion for every real number, i.e. 0.5 = 0.49999··· ) Definition: Given any two sets A and B, we say that card(A) ≤ card(B) if there is a injection from A into B. Schröder-Bernstein Theorem Given two sets A and B, if card(A) ≤ card(B) and card(A) ≥ card(B), then card(A) = card(B). Consequence of the Schröder-Bernstein Theorem card (2ω ) = card(10ω ) In other words, the set of infinite binary sequences has the same cardinality as the set of infinite decimal sequences. A binary sequence can only use the digits 0 and 1, an example is {1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 0, … } A decimal sequence can use the ten digits 0 to 9, an example is {2, 5, 6, 0, 4, 3, 3, 7, 8, 4, 8, 9, 1, 2, … } Theorem The set of real numbers has the same cardinality as the set S = { x : 0 < x < 1} Proof: Consider the function f ( x) tan( π π x ) 2 2 which is one-to-one and maps (0, 1) exactly onto (-∞, ∞). Countable Sets Definition: A set S is said to be countably infinite if it has the same cardinality as the set of natural numbers N. A set S is said to be countable if it is either finite or countably infinite. Loosely speaking, a set is countable if you can put its elements in a linear order such that every elements has only finitely many predecessors. A set S is said to be uncountable if it is not countable. Theorem (1) Any subset of a countable set is countable. (2) The union of two countable sets is countable. (3) The Cartesian product of any two countable sets is countable. Examples: (a) The set Z of integers is countable. (b) The set Q of rational numbers is countable. Axiom of countable choice Any countable union of countable sets is countable. (Note: this axiom cannot be proved by ZF.) Theorem The set Q+ of positive rational numbers is countable. Proof: 1 , 1 2 , 1 3 , 1 4 , 1 5 , 1 6 , 1 1 , 2 2 , 2 3 , 2 4 , 2 5 , 2 6 , 2 1 , 3 2 , 3 3 , 3 4 , 3 5 , 3 6 , 3 1 , 4 2 , 4 3 , 4 4 , 4 5 , 4 6 , 4 1 , 5 2 , 5 3 , 5 4 , 5 5 , 5 6 , 5 1 , 6 2 , 6 3 , 6 4 , 6 5 , 6 6 , 6 1 , 7 2 , 7 3 , 7 4 , 7 5 , 7 6 , 7 Theorem The set of real numbers in the interval [0, 1] is uncountable. Proof: (Cantor Diagonalization method) By contradiction. Suppose that we can arrange those numbers in a sequence {a1, a2, a3, } and since each number in the sequence is a decimal, we can write their decimal forms as a1 = 0.a11a12a13a14… a2 = 0.a21a22a23a24… a3 = 0.a31a32a33a34… … (here we use the non-terminating form if there are two choices, i.e. 1 = 0.999…) Next we construct a number b whose decimal form is b = 0.b1b2b3b4… and such that bk = akk + 1 if akk < 9 and bk = 0 if akk = 9 Then b will be a real number between 0 and 1, but b is not equal to any of the numbers in the list {a1, a2, a3, …}, a contradiction! Definition The first infinite ordinal number ω is defined to be the set {0, 1, 2, 3, } ω +1 is defined to be the 2nd infinite ordinal number, and it is the set {0, 1, 2, 3, , ω} (note: there is no ω – 1 because ω is not constructed by adding just one more element to any set.) ω + (n +1) is the set {0, 1, 2, 3, , ω, ω+1, ω+2, , ω+n} ω + ω is the set {0, 1, 2, 3, , ω, ω+1, ω+2, ω+3, } Theorem ω + n is countable for any whole number n. ω + ω is countable. ω2 (= ω×ω) is countable. ωn is countable for any whole number n. ωω is uncountable. Definition 2ω is the set of all functions from ω to 2 (={0, 1}) ωω is the set of all functions from ω to ω. Theorem (a) card(2ω) = card((ω)) (b) card(2ω) = card(ωω) (c) card() = card(2ω) Old Theorem Revisit Theorem Given any set S, the power set (S) has a strictly larger cardinality than S. This implies that we can construct sets with larger and larger cardinalities using power sets: S, (S), ((S)), … Comment Since any set obtained from ω using set operations other than power set is countable, Cantor conjectured the following Continuum Hypothesis There is no set whose cardinality is strictly between that of card(ω) and card((ω)) It turns out that this hypothesis is independent of the ZFC axioms. (Paul Cohen, 1963) General Continuum Hypothesis For any infinite set X, there is no set whose cardinality is strictly between that of card(X) and card((X)) Application in computer science Theorem Given any computer language, the set of programs in that language is countable. Corollary The set of computable functions from ω to ω is countable. But there are uncountably many functions from ω to ω. Hence there are uncountably many functions from ω to ω that are not computable (i.e. they cannot be generated by a computer program.) This means computers can never take over the jobs of human.