Discrete Mathematics and its Applications 9/8/2019 University of Aberdeen, Computing Science CS3511 Discrete Methods Kees van Deemter Slides adapted from Michael P. Frank’s Course Based on the Text Discrete Mathematics & Its Applications (5th Edition) by Kenneth H. Rosen A word about organization: Since different courses have different lengths of lecture periods, and different instructors go at different paces, rather than dividing the material up into fixed-length lectures, we will divide it up into “modules” which correspond to major topic areas and will generally take 1-3 lectures to cover. Within modules, we have smaller “topics”. Within topics are individual slides. The instructor can bring several modules to each lecture with him, to make sure he has enough material to fill the lecture, or in case he wants to preview or review slides from upcoming or recent past lectures. 9/8/2019 Michael P. Frank / Kees van Deemter (c)2001-2002, Michael P. Frank

Rosen 5th ed., §1.8 ~60 slides, ~1.5 lectures Module #4: Functions Rosen 5th ed., §1.8 ~60 slides, ~1.5 lectures 9/8/2019 Michael P. Frank / Kees van Deemter

Michael P. Frank / Kees van Deemter Functions From calculus, you know the concept of a real-valued function f, which assigns to each number xR one particular value y=f(x), where yR. Example: f defined by the rule f(x)=x2 The notion of a function can be generalized to the concept of assigning elements of any set to elements of any set. Functions are also called operators. 9/8/2019 Michael P. Frank / Kees van Deemter

Function: Formal Definition A function f from (or “mapping”) A to B (f:AB) is an assignment of exactly one element f(x)B to each element xA. Some further generalizations of this idea: Functions of n arguments: f: (A1 x A2... x An)  B. A partial (non-total) function f assigns zero or one elements of B to each element xA. 9/8/2019 Michael P. Frank / Kees van Deemter

Michael P. Frank / Kees van Deemter Graphs of Functions We can represent a function f:AB as a set of ordered pairs f ={(a,f(a)) | aA}. This makes f a relation between A and B: f is a subset of A x B. But functions are special: for every aA, there is at least one pair (a,b). Formally: aAbB((a,b)f) for every aA, there is at most one pair (a,b). Formally: a,b,c((a,b)f  (a,c)f  bc) A relation over numbers can be represent as a set of points on a plane. (A point is a pair (x,y).) A function is then a curve (set of points), with only one y for each x. 9/8/2019 Michael P. Frank / Kees van Deemter

Discrete Mathematics and its Applications 9/8/2019 Functions can be represented graphically in several ways: A B f • • f • • • • y • a b • Note that the inverses of the functions in the two diagrams on the left are not functions. Note also that the plot has gaps if the domain of the function is not the set R • • • x A B Bipartite Graph Plot Like Venn diagrams 9/8/2019 Michael P. Frank / Kees van Deemter (c)2001-2002, Michael P. Frank

Functions We’ve Seen So Far A proposition might be viewed as a function from “situations” to truth values {T,F} p=“It is raining.” s=our situation here,now p(s){T,F}. A propositional operator can be viewed as a function from ordered pairs of truth values to truth values: e.g., ((F,T)) = T. Another example: →((T,F)) = F. 9/8/2019 Michael P. Frank / Kees van Deemter

Michael P. Frank / Kees van Deemter More functions so far… A predicate (or property)can be viewed as a function from objects to propositions: P :≡ “is 7 feet tall”; P(Mike) = “Mike is 7 feet tall.” A set S over universe U can be viewed as a function from the elements of U to ... 9/8/2019 Michael P. Frank / Kees van Deemter

Michael P. Frank / Kees van Deemter Still More Functions A set S over universe U can be viewed as a function from the elements of U to … … {T, F}, saying for each element of U whether it is in S. (This is called the characteristic function of S) Suppose U={0,1,2,3,4}. Then S={1,3} S(0)=S(2)=S(4)=F, S(1)=S(3)=T. 9/8/2019 Michael P. Frank / Kees van Deemter

Michael P. Frank / Kees van Deemter Still More Functions A set operator, such as  or , can be viewed as a function from … to … 9/8/2019 Michael P. Frank / Kees van Deemter

Michael P. Frank / Kees van Deemter Still More Functions A set operator such as  or  can be viewed as a function … … from (ordered) pairs of sets, to sets. Example: (({1,3},{3,4})) = {3} 9/8/2019 Michael P. Frank / Kees van Deemter

Michael P. Frank / Kees van Deemter A new notation Sometimes we write YX to denote the set F of all possible functions f: XY. Thus, f  YX is another way of saying that f: XY. (This notation is especially appropriate, because for finite X, Y, we have |F| = |Y||X|. ) 9/8/2019 Michael P. Frank / Kees van Deemter

A Trick that’s sometimes used If we use representations F0, T1, then a subset TS is a function from S to {0,1}, Therefore, P(S) can be represented as {0,1}S (the set of all functions from S to {0,1} ) (Note that if we also represent 2:{0,1}, then P(S) can be represented as 2S . The powerset of S is sometimes written this way to stress the cardinality of the powerset.) 9/8/2019 Michael P. Frank / Kees van Deemter

Some Function Terminology If f:AB, and f(a)=b (where aA & bB), then we say: A is the domain of f. B is the codomain of f. b is the image of a under f. a is a pre-image of b under f. In general, b may have more than 1 pre-image. The range RB of f is R={b | a f(a)=b }. We also say the signature of f is A→B. 9/8/2019 Michael P. Frank / Kees van Deemter

Michael P. Frank / Kees van Deemter Range versus Codomain The range of a function may not be its whole codomain. The codomain is the set that the function is declared to map all domain values into. The range is the particular set of values in the codomain that the function actually maps elements of the domain to. (One might say: The range is the smallest set that could be used as its codomain.) 9/8/2019 Michael P. Frank / Kees van Deemter

Range vs. Codomain - Example Suppose I declare to you that: “f is a function mapping students in this class to the set of grades {A,B,C,D,E}.” At this point, you know f’s codomain is: __________, and its range is ________. Suppose the grades turn out all As and Bs. Then the range of f is _________, but its codomain is __________________. {A,B,C,D,E} unknown {A,B} still {A,B,C,D,E} 9/8/2019 Michael P. Frank / Kees van Deemter

(n-ary) functions on a set An n-ary function (also: n-ary operator) over (also: on) S is any function from the set of ordered n-tuples of elements of S, to S itself. E.g., if S={T,F},  can be seen as a unary operator, and , are binary operators on S. Another example:  and  are binary operators on the set of all sets. 9/8/2019 Michael P. Frank / Kees van Deemter

Operators over functions If  (“dot”) is an n-ary operator over B, then we can extend  to also denote an operator over functions from A to B. E.g.: Given any binary operator :BBB, and functions f,g:AB, we define (f  g):AB to be the function defined by: aA, (f  g)(a) = f(a)g(a). 9/8/2019 Michael P. Frank / Kees van Deemter

Function Operator Example ,× (plus,times) are binary operators over R. (Normal addition & multiplication.) Therefore, we can also “add” and “multiply” functions f,g:RR: (f  g):RR, where (f  g)(x) = f(x)  g(x) (f × g):RR, where (f × g)(x) = f(x) × g(x) 9/8/2019 Michael P. Frank / Kees van Deemter

Function Composition Operator Note match here. For functions g:AB and f:BC, there is a special operator called compose (“○”). It composes (creates) a new function out of f and g by applying f to the result of applying g. We say (f○g):AC, where (f○g)(a) :≡ f(g(a)). g(a)B, so f(g(a)) is defined and f(g(a))C. Note that ○ is non-commutative (i.e., we don’t always have f○g = g○f). 9/8/2019 Michael P. Frank / Kees van Deemter

Function Composition Operator “We don’t always have f○g = g○f “ Can you express this in predicate logic? 9/8/2019 Michael P. Frank / Kees van Deemter

Function Composition Operator Discrete Mathematics and its Applications 9/8/2019 Function Composition Operator “We don’t always have f○g = g○f “ Can you express this in predicate logic? (fgx(f○g(x) =g○f (x))). [Do not write: fgx(f○g(x)  g○f (x)))] (Note that this formula quantifies over functions as well as ordinary objects – something that is not possible in First Order Predicate Logic (FOPL), which is what was taught earlier in this course.) Two functions whose composition commutes for all their arguments are f(x)=x^2 and g(x)=square root of x. Two functions whose composition never commutes: f(x)=x^2 and g(x)=-x. 9/8/2019 Michael P. Frank / Kees van Deemter (c)2001-2002, Michael P. Frank

Images of Sets under Functions Given f:AB, and SA, The image of S under f is the set of all images (under f) of the elements of S. f(S) : {f(s) | sS} : {b | sS: f(s)=b}. The range of f equals the image (under f) of ... 9/8/2019 Michael P. Frank / Kees van Deemter

Images of Sets under Functions Given f:AB, and SA, The image of S under f is the set of all images (under f) of the elements of S. f(S) : {f(s) | sS} : {b | sS: f(s)=b}. The range of f equals the image (under f) of f’s domain. 9/8/2019 Michael P. Frank / Kees van Deemter

Michael P. Frank / Kees van Deemter One-to-One Functions A function is one-to-one (1-1), or injective, or an injection, iff every element of its range has only 1 pre-image. Formally: given f:AB, “x is injective” : (x,y: xy  f(x)f(y)). In other words: only one element of the domain is mapped to any given one element of the range. In this case, domain & range have same cardinality. What about codomain? 9/8/2019 Michael P. Frank / Kees van Deemter

Michael P. Frank / Kees van Deemter Codomain may be larger. 9/8/2019 Michael P. Frank / Kees van Deemter

One-to-One Illustration Are these relations one-to-one functions? • • • • • • • • • • • • • • • • • • • • • • • • • • • 9/8/2019 Michael P. Frank / Kees van Deemter

One-to-One Illustration Are these relations one-to-one functions? • • • • • • • • • • • • • • • • • • • • • • • • • • • One-to-one 9/8/2019 Michael P. Frank / Kees van Deemter

One-to-One Illustration Are these relations one-to-one functions? • • • • • • • • • • • • • • • • • • • • • • • • • • • Not one-to-one One-to-one 9/8/2019 Michael P. Frank / Kees van Deemter

One-to-One Illustration Are these relations one-to-one functions? • • • • • • • • • • • • • • • • • • • • • • • • • • • Not one-to-one Not even a function! One-to-one 9/8/2019 Michael P. Frank / Kees van Deemter

Sufficient Conditions for 1-1ness For functions f over numbers, we say: f is strictly increasing iff x>y  f(x)>f(y) for all x,y in domain; f is strictly decreasing iff x>y  f(x)<f(y) for all x,y in domain; If f is either strictly increasing or strictly decreasing, then f must be one-to-one. Does the converse hold? 9/8/2019 Michael P. Frank / Kees van Deemter

Michael P. Frank / Kees van Deemter If f is either strictly increasing or strictly decreasing, then f is one-to-one. Does the converse hold? 9/8/2019 Michael P. Frank / Kees van Deemter

Michael P. Frank / Kees van Deemter If f is either strictly increasing or strictly decreasing, then f is one-to-one. Does the converse hold? NO E.g., f:NN such that if x is even then f(x)=x+1 if x is odd then f(x)=x-1 9/8/2019 Michael P. Frank / Kees van Deemter

Onto (Surjective) Functions A function f:AB is onto or surjective or a surjection iff its range is equal to its codomain (bB, aA: f(a)=b). Consider “country of birth of”: AB, where A=people, B=countries. Is this a function? Is it an injection? Is it a surjection? 9/8/2019 Michael P. Frank / Kees van Deemter

Onto (Surjective) Functions A function f:AB is onto or surjective or a surjection iff its range is equal to its codomain Consider “country of birth of”: AB, where A=people, B=countries. Is this a function? Yes (always 1 c.o.b.) Is it an injection? No (many have same c.o.b.) Is it a surjection? Probably yes .. 9/8/2019 Michael P. Frank / Kees van Deemter

Onto (Surjective) Functions A function f:AB is onto or surjective or a surjection iff its range is equal to its codomain In predicate logic: 9/8/2019 Michael P. Frank / Kees van Deemter

Onto (Surjective) Functions A function f:AB is onto or surjective or a surjection iff its range is equal to its codomain. In predicate logic: bBaA f(a)=b 9/8/2019 Michael P. Frank / Kees van Deemter

Onto (Surjective) Functions A function f:AB is onto or surjective or a surjection iff its range is equal to its codomain (bBaA f(a)=b). Think: An onto function maps the set A onto (over, covering) the entirety of the set B, not just over a piece of it. E.g., for domain & codomain R, x3 is onto, whereas x2 isn’t. (Why not?) 9/8/2019 Michael P. Frank / Kees van Deemter

Onto (Surjective) Functions E.g., for domain & codomain R, x3 is onto, but x2 is not. (Why not?) Consider f:RR such that, for all x, f(x)=x2. Consider any negative number a=-b in R. x(x2=a). So f is not surjective. Consider f:RR such that for all x, f(x)=x3. Consider any negative number a=-b in R. Let z be such that z3=b. Then (-z)3=-b=a 9/8/2019 Michael P. Frank / Kees van Deemter

Michael P. Frank / Kees van Deemter The Identity Function For any domain A, the identity function I:AA (also written IA) on A is the function such that everything is mapped to itself In predicate logic: 9/8/2019 Michael P. Frank / Kees van Deemter

Michael P. Frank / Kees van Deemter The Identity Function For any domain A, the identity function I:AA (also written IA) on A is the function such that everything is mapped to itself In predicate logic: aA I(a)=a. 9/8/2019 Michael P. Frank / Kees van Deemter

Michael P. Frank / Kees van Deemter The Identity Function For any domain A, the identity function I:AA (also written IA) on A is the function such that aA I(a)=a. Is the identity function one-to-one (injective)? onto (surjective)? 9/8/2019 Michael P. Frank / Kees van Deemter

Michael P. Frank / Kees van Deemter The Identity Function For any domain A, the identity function I:AA (also written IA) on A is the function such that aA I(a)=a. Is the identity function one-to-one (injective)? Yes onto (surjective)? Yes 9/8/2019 Michael P. Frank / Kees van Deemter

Identity Function Illustrations The identity function: • • • y y = I(x) = x • • • • • • x Domain and range 9/8/2019 Michael P. Frank / Kees van Deemter

Michael P. Frank / Kees van Deemter Illustration of Onto Are these functions onto their depicted co-domains? • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 9/8/2019 Michael P. Frank / Kees van Deemter

Michael P. Frank / Kees van Deemter Illustration of Onto Are these functions onto? • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 9/8/2019 Michael P. Frank / Kees van Deemter

Michael P. Frank / Kees van Deemter Illustration of Onto Are these functions onto? • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • onto not onto onto not onto 9/8/2019 Michael P. Frank / Kees van Deemter

Michael P. Frank / Kees van Deemter Illustration of Onto Are these functions 1-1? • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • onto not onto onto not onto 9/8/2019 Michael P. Frank / Kees van Deemter

Michael P. Frank / Kees van Deemter Illustration of Onto Are these functions 1-1? • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • not 1-1 onto not 1-1 not onto 1-1 onto 1-1 not onto 9/8/2019 Michael P. Frank / Kees van Deemter

Michael P. Frank / Kees van Deemter Bijections A function is said to be a one-to-one correspondence, or a bijection iff it is both one-to-one and onto. 9/8/2019 Michael P. Frank / Kees van Deemter

Two terminologies for talking about functions injection = one-to-one surjection = onto bijection = one-to-one correspondence 3 = 1&2 9/8/2019 Michael P. Frank / Kees van Deemter

Michael P. Frank / Kees van Deemter Bijections For bijections f:AB, there exists an inverse of f, written f 1: BA Intuitively, this is the function that undoes everything that f does Formally, it’s the unique function such that ... 9/8/2019 Michael P. Frank / Kees van Deemter

Michael P. Frank / Kees van Deemter Bijections For bijections f:AB, there exists an inverse of f, written f 1: BA Intuitively, this is the function that undoes everything that f does Formally, it’s the unique function such that (recall that IA is the identity function on A) 9/8/2019 Michael P. Frank / Kees van Deemter

Michael P. Frank / Kees van Deemter Bijections Example 1: Let f: ZZ be defined as f(x)=x+1. What is f1 ? Example 2: Let g: ZN be defined as g(x)=|x|. What is g1 ? 9/8/2019 Michael P. Frank / Kees van Deemter

Michael P. Frank / Kees van Deemter Bijections Example 1: Let f: ZZ be defined as f(x)=x+1. What is f1 ? f1 is the function (let’s call it h) h: ZZ defined as h(x)=x-1. Proof: h(f(x)) = (x+1)-1 = x 9/8/2019 Michael P. Frank / Kees van Deemter

Michael P. Frank / Kees van Deemter Bijections Example 2: Let g: ZN be defined as g(x)=|x|. What is g1 ? This was a trick question: there is no such function, since g is not a bijection: There is no function h such that h(|x|)=x and h(|x|)=x (NB There is a relation h for which this is true.) 9/8/2019 Michael P. Frank / Kees van Deemter

Aside About Representations It is possible to represent any type of discrete structure (propositions, bit-strings, numbers, sets, ordered pairs, functions) in terms of some combination of other structures. Perhaps none of these structures is more fundamental than the others. However, logic, and sets are often used as the foundation for all else. E.g. in  9/8/2019 Michael P. Frank / Kees van Deemter

A Couple of Key Functions In discrete math, we frequently use the following two functions over real numbers: The floor function ·:R→Z, where x (“floor of x”) means the largest integer  x. I.e., x :≡ max({iZ|i≤x}). The ceiling function · :R→Z, where x (“ceiling of x”) means the smallest integer  x. x :≡ min({iZ|i≥x}) 9/8/2019 Michael P. Frank / Kees van Deemter

Visualizing Floor & Ceiling Discrete Mathematics and its Applications 9/8/2019 Visualizing Floor & Ceiling Real numbers “fall to their floor” or “rise to their ceiling.” 3 . 1.6=2 2 . 1.6 . 1 1.6=1 . 1.4= 1 In other words, it matters whether you first apply minus then apply ceiling/floor, or the other way round (i.e. first apply ceiling/floor then apply minus). 1 . 1.4 . 2 1.4= 2 . . . 3 3 3=3= 3 9/8/2019 Michael P. Frank / Kees van Deemter (c)2001-2002, Michael P. Frank

Do these equalities hold? x =  x & x =  x 9/8/2019 Michael P. Frank / Kees van Deemter

It depends on whether x is an integer If xZ then x = x = x, so x = x =  x & x = x =  x But if xZ, then x   x & x   x E.g., 3.4 = 4  3 =  3.4 9/8/2019 Michael P. Frank / Kees van Deemter

Plots with floor/ceiling Note that for f(x)=x, the graph of f includes the point (a, 0) for all values of a such that a0 and a<1, but not for the value a=1. {xR : x = 0} = (informally) = {0,..,0.1,..0.2,..,…0.9,…} does not include its limit 1. Sets that do not include all of their limit points are generally called open sets. In a plot, we draw a limit point of a curve using an open dot (circle) if the limit point is not on the curve, and with a closed (solid) dot if it is on the curve. 9/8/2019 Michael P. Frank / Kees van Deemter

Plots with floor/ceiling: Example Plot of graph of function f(x) = x/3: f(x) Set of points (x, f(x)) +2 3 x +3 2 9/8/2019 Michael P. Frank / Kees van Deemter

Review of §1.8 (Functions) Function variables f, g, h, … Notations: f:AB, f(a), f(A). Terms: image, pre-image, domain, codomain, range, one-to-one, onto, strictly (in/de)creasing, bijective, inverse, composition. Function unary operator f 1, binary operators , , etc., and ○. The RZ functions x and x. 9/8/2019 Michael P. Frank / Kees van Deemter