Discrete Mathematics R. Johnsonbaugh Chapter 2 The Language of Mathematics

2.1 Sets Set = a collection of distinct unordered objects Members of a set are called elements How to determine a set Listing: Example: A = {1,3,5,7} = {7, 5, 3, 1, 3} Description Example: B = {x | x = 2k + 1, 0 < k < 30}

Finite and infinite sets Examples: A = {1, 2, 3, 4} B = {x | x is an integer, 1 < x < 4} Infinite sets Z = {integers} = {…, -3, -2, -1, 0, 1, 2, 3,…} S={x| x is a real number and 1 < x < 4} = [0, 4]

Some important sets The empty set  = { } has no elements. Also called null set or void set. Universal set: the set of all elements about which we make assertions. Examples: U = {all natural numbers} U = {all real numbers} U = {x| x is a natural number and 1< x<10}

Cardinality Cardinality of a set A (in symbols |A|) is the number of elements in A Examples: If A = {1, 2, 3} then |A| = 3 If B = {x | x is a natural number and 1< x< 9} then |B| = 9 Infinite cardinality Countable (e.g., natural numbers, integers) Uncountable (e.g., real numbers)

Subsets X is a subset of Y if every element of X is also contained in Y (in symbols X  Y) Equality: X = Y if X  Y and Y  X, i.e., X = Y whenever x  X, then x  Y, and whenever x  X, then x  X X is a proper subset of Y if X  Y but Y  X Observation:  is a subset of every set

Power set The power set of X is the set of all subsets of X, in symbols P(X),i.e. P(X)= {A | A  X} Example: if X = {1, 2, 3}, then P(X) = {, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}} Theorem 2.1.4: If |X| = n, then |P(X)| = 2n. See proof by induction in textbook

Set operations: Union and Intersection Given two sets X and Y The union of X and Y is defined as the set X  Y = { x | x  X or x  Y} The intersection of X and Y is defined as the set X  Y = { x | x  X and x  Y} Two sets X and Y are disjoint if X  Y = 

Complement and Difference The difference of two sets X – Y = { x | x  X and x  Y} The difference is also called the relative complement of Y in X Symmetric difference X Δ Y = (X – Y)  (Y – X) The set of all elements that belong to X or to Y but not both X and Y. The complement of a set A contained in a universal set U is the set Ac = U – A In symbols Ac = U - A

Example If X={1, 4, 7, 10}, Y={1, 2, 3, 4, 5} X  Y = X  Y = X – Y = X Δ Y = (how else can you write this?)

Example If X={1, 4, 7, 10}, Y={1, 2, 3, 4, 5} X  Y = {1, 2, 3, 4, 5, 7, 10} X  Y = {1, 4} X – Y = {7, 10} Y – X = {2, 3, 5} X Δ Y = (X – Y)  (Y – X) = {2, 3, 5, 7, 10}

Venn diagrams A Venn diagram provides a graphic view of sets Set union, intersection, difference, symmetric difference and complements can be easily and visually identified U C A B

Properties of set operations Theorem 2.1.10: Let U be a universal set, and A, B and C subsets of U. The following properties hold: Associativity: (A  B)  C = A  (B  C) (A  B)  C = A  (B C) Commutativity: A  B = B  A A  B = B  A

Properties of set operations Distributive laws: A(BC) = (AB)(AC) A(BC) = (AB)(AC) Identity laws: AU=A A = A Complement laws: AAc = U AAc = 

Properties of set operations Idempotent laws: AA = A AA = A Bound laws: AU = U A =  Absorption laws: A(AB) = A A(AB) = A

Properties of set operations Involution law: (Ac)c = A 0/1 laws: c = U Uc =  De Morgan’s laws for sets: (AB)c = AcBc (AB)c = AcBc

Addition Principle A.K.A The Inclusion-Exclusion Principle If A and B are finite sets then, | A  B | = |A| + |B| - | A  B | U A  B A B

Addition Principle for Disjoint Sets | A  B  C | = |A| + |B| + |C| - |A  B| - |B  C| - |A  C| + |A  B  C| A = { a, b, c, d, e } B = { a, b, e, g, h } C = { b, d, e, g, h, k, m, n} A company wants to hire 25 programmers to handle systems Programming jobs and 40 programmers for applications programming. Of those hired, ten will be expected to perform jobs of both types. How many programmers must be hired?

One more example A survey was taken on methods of commuter travel. Each respondent was asked to check BUS, TRAIN, or CAR as a major mode of traveling. More than one answer is allowed. The results are: BUS 30 TRAIN 35 CAR 100 BUS and TRAIN 15 BUS and CAR 15 TRAIN and CAR 20 All three modes 5 How many people drank coffee while traveling? How many people completed a survey form

2.2 Functions A function f from X to Y (in symbols f : X  Y) is a relation from X to Y such that Dom(f) = X and if two pairs (x,y) and (x,y’)  f, then y = y’ Example: Dom(f) = X = {a, b, c, d}, Rng(f) = {1, 3, 5} f(a) = f(b) = 3, f(c) = 5, f(d) = 1.

Domain and Range Domain of f = X Range of f = { y | y = f(x) for some x X} A function f : X  Y assigns to each x in Dom(f) = X a unique element y in Rng(f)  Y. Therefore, no two pairs in f have the same first coordinate.

Algebraically speaking Note that such definitions on functions are consistent with what you have seen in your Calculus courses. function not a function  1 intersection  violations when > 1

Modulus operator Let x be a nonnegative integer and y a positive integer r = x mod y is the remainder when x is divided by y Examples: 1 = 13 mod 3 6 = 234 mod 19 4 = 2002 mod 111 Basically, remove the complete y’s and count what’s left mod is called the modulus operator

One-to-one functions A function f : X  Y is one-to-one  for each y  Y there exists at most one x  X with f(x) = y. (therefore, f(x) = c is out of play) Alternative definition: f : X  Y is one-to-one  for each pair of distinct elements x1, x2  X there exist two distinct elements y1, y2  Y such that f(x1) = y1 and f(x2) = y2. Examples: 1. The function f(x) = 2x from the set of real numbers to itself is one-to-one 2. The function f : R  R defined by f(x) = x2 is not one-to-one, since for every real number x, f(x) = f(-x).

Onto functions A function f : X  Y is onto (or, subjective)  for each y  Y there exists at least one x  X with f(x) = y, i.e. Rng(f) = Y. Example: The function f={1,a),(2,c),(3,b)} from X={1,2,3} to Y={a,b,c} is 1-to-1 and onto. If Y={a,b,c,d}, then still 1-to-1, but not onto. Example: The function f(x) = ex from the set of real numbers to itself is not onto Y = the set of all real numbers. However, if Y is restricted to Rng(f) = R +, the set of positive real numbers, then f(x) is onto. Why? Look at the several visual examples illustrated in the textbook

Bijective functions A function f : X Y is bijective  f is one-to-one and onto Examples: 1. Is A linear function f(x) = ax + b a bijective function from the set of real numbers to itself. Why? 2. Is the function f(x) = x3 a bijective from the set of real numbers to itself. Why?

Inverse function Given a function y = f(x), the inverse f -1 is the set {(y, x) | y = f(x)} The inverse f -1 of f is not necessarily a function Example: if f(x) = x2, then f -1 (4) = 4 = ± 2, not a unique value and therefore f is not a function However, if f is a bijective function, it can be shown that f -1 is a function See Example 2.2.35.

Exponential and logarithmic functions Let f(x) = 2x and g(x) = log 2 x = lg x f ◦ g(x) = f(g(x)) = f(lg x) = 2 lg x = x g ◦ f(x) = g(f(x)) = g(2x) = lg 2x = x Exponential and logarithmic functions are inverse functions

Composition of functions Given two functions g : X  Y and f : Y  Z, the composition f ◦ g is defined as follows: f ◦ g (x) = f(g(x)) for every x  X. Example: g(x) = x2 –1 f(x) = 3x + 5 g ◦ f(x) = g(f(x)) = g(3x + 5) = (3x + 5)2 - 1 Composition of functions is associative f ◦ (g ◦h) = (f ◦ g) ◦ h In general, it is not commutative f ◦ g  g ◦ f.

Binary operators A binary operator on a set X is a function f that associates a single element of X to every pair of elements in X, i.e. f : X x X  X and f(x1, x2)  X for every pair of elements x1, x2. Examples of binary operators are addition,subtraction and multiplication of real numbers, taking unions or intersections of sets, concatenation of two strings over a set X, etc.

Unary operators A unary operator on a set X associates to each single element of X one element of X. Examples: Let X = U be a universal set and P(U) the power set of U Define f : P(U)  P(U) the function defined by f (A) = A‘ the set complement of A in U, for every A  U. Then f defines a unary operator on P(U). (The operator here is the “complement” itself).

2.3 Sequences and strings A sequence is an ordered list of numbers, usually defined according to a formula function, sn, n = 1, 2, 3,... is the index of the sequence If s is a sequence {sn| n = 1, 2, 3,…}, s1 denotes the first element, s2 the second element,… sn the nth element… {n} is called the indexing set of the sequence. Usually the indexing set is N (natural numbers) or an infinite subset of N.

Examples of sequences Let s = {sn} be the sequence defined by sn = 1/n , for n = 1, 2, 3,… The first few elements of the sequence are: 1, ½, 1/3, ¼, 1/5,1/6,… sn = n2 + 1, for n = 1, 2, 3,… The first few elements of s are: 2, 5, 10, 17, 26, 37, 50,…

Increasing and decreasing A sequence s = {sn} is said to be increasing if sn < sn+1 decreasing if sn > sn+1, for every n = 1, 2, 3,… Examples: Sn = 4 – 2n, n = 1, 2, 3,… is decreasing: 2, 0, -2, -4, -6,… Sn = 2n -1, n = 1, 2, 3,… is increasing: 1, 3, 5, 7, 9, …

Subsequences A subsequence of a sequence s = {sn} is a sequence t = {tn} that consists of certain elements of s retained in the original order they had in s Example: let s = {sn = n | n = 1, 2, 3,…} 1, 2, 3, 4, 5, 6, 7, 8,… Let t = {tn = 2n | n = 1, 2, 3,…} 2, 4, 6, 8, 10, 12, 14, 16,… t is a subsequence of s

Sigma notation  ak = a1 + a2 + … + am If {an} is a sequence, then the sum m  ak = a1 + a2 + … + am k = 1 This is called the “sigma notation”, where the Greek letter  indicates a sum of terms from the sequence

Pi notation  ak = a1a2…am If {an} is a sequence, then the product m This is called the “pi notation”, where the Greek letter  indicates a product of terms of the sequence

Strings Let X be a nonempty set. A string over X is a finite sequence of elements from X. Example: if X = {a, b, c} Then  = bbaccc is a string over X Notation: bbaccc = b2ac3 The length of a string  is the number of elements of  and is denoted by ||. If  = b2ac3 then || = 6 The null string is the string with no elements and is denoted by the Greek letter  (lambda). It has length zero.

More on strings Let X* = {all strings over X including } Let X+ = X* - {}, the set of all non-null strings Concatenation of two strings  and  is the operation on strings consisting of writing  followed by  to produce a new string  Example:  = bbaccc and  = caaba, then  = bbaccccaaba = b2ac4a2ba Clearly, || = | | + ||