R. Johnsonbaugh, Discrete Mathematics 5 th edition, 2001 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} Description  Example: B = {x | x = 2k + 1, 0 < k < 3}

Finite and infinite sets  Finite sets Examples:  A = {1, 2, 3, 4}  B = {x | x is an integer, 1 < x < 4}  Infinite sets  Examples:  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  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)| = 2 n.

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 complement of a set A contained in a universal set U is the set A c = U – A In symbols A c = U - A

Venn diagrams  A Venn diagram provides a graphic view of sets  Set union, intersection, difference, symmetric difference and complements can be identified

Properties of set operations (1) Theorem : Let U be a universal set, and A, B and C subsets of U. The following properties hold: a) Associativity: (A  B)  C = A  (B  C) (A  B)  C = A  (B  C) b) Commutativity: A  B = B  A A  B = B  A

Properties of set operations (2) c) Distributive laws: A  (B  C) = (A  B)  (A  C) A  (B  C) = (A  B)  (A  C) d) Identity laws: A  U=A A  = A e) Complement laws: A  A c = U A  A c = 

Properties of set operations (3) f) Idempotent laws: A  A = A A  A = A g) Bound laws: A  U = U A  =  h) Absorption laws: A  (A  B) = A A  (A  B) = A

Properties of set operations (4) i) Involution law: (A c ) c = A j) 0/1 laws:  c = U U c =  k) De Morgan’s laws for sets: (A  B) c = A c  B c (A  B) c = A c  B c

2.2 Sequences and strings  A sequence is an ordered list of numbers, usually defined according to a formula: s n = a function of n = 1, 2, 3,...  If s is a sequence {s n | n = 1, 2, 3,…}, s 1 denotes the first element, s 2 the second element,… s n 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 Examples: 1. Let s = {s n } be the sequence defined by s n = 1/n, for n = 1, 2, 3,… The first few elements of the sequence are: 1, ½, 1/3, ¼, 1/5,1/6,… 2. Let s = {s n } be the sequence defined by s n = n 2 + 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 = {s n } is said to be increasing if s n < s n+1 decreasing is s n > s n+1, for every n = 1, 2, 3,… Examples: S n = 4 – 2n, n = 1, 2, 3,… is decreasing: 2, 0, -2, -4, -6,… S n = 2n -1, n = 1, 2, 3,… is increasing: 1, 3, 5, 7, 9, …

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

Sigma notation If {a n } is a sequence, then the sum m  a k = a 1 + a 2 + … + a m k = 1 This is called the “sigma notation”, where the Greek letter  indicates a sum of terms from the sequence

Pi notation If {a n } is a sequence, then the product m  a k = a 1 a 2 …a m k=1 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 = b 2 ac 3 The length of a string  is the number of elements of  and is denoted by |  |. If  = b 2 ac 3 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 = b 2 ac 4 a 2 ba Clearly, |  | = |  | + |  |

2.3 Number systems  Binary digits: 0 and 1, called bits.  In this section we study: binary, hexadecimal and octal number systems. Review of decimal system: Example: 45,238 is equal to 8ones8 x 1 =8 3tens3 x 10 = 30 2 hundreds2 x 100 = 200 5thousands5 x 1000 = ten thousands4 x = 40000

Binary number system  From binary to decimal:  The number is equivalent to 1 one1 x2 0 =1 1 two1x2 1 = 2 0 four0x2 2 =0 1 eight1x2 3 =8 0 sixteen0x2 4 = 0 1 thirty-two1x2 5 = 32 1 sixty-four1x2 6 = in decimal base

From decimal to binary  The number is equivalent to 73 = 2 x 36 + remainder 1 36 = 2 x 18 + remainder 0 18 = 2 x 9 + remainder 0 9 = 2 x 4 + remainder 1 4 = 2 x 2 + remainder 0 2 = 2 x 1 + remainder 0  = (write the remainders in reverse order preceded by the quotient)

Binary addition table 

Adding binary numbers  Example: add  carry ones

Hexadecimal number system Decimal system ABCDEF Hexadecimal system

Hexadecimal to decimal  The hexadecimal number 3A0B 16 is 11 x 16 0 =11 0 x 16 1 = 0 10 x 16 2 = x 16 3 =

Decimal to hexadecimal Given the number = 146x16 + remainder = 9x16 + remainder is equivalent to the hexadecimal number

Hexadecimal addition Add 23A F 16 23A F 16 2C9 16

2.4 Relations  Given two sets X and B, its Cartesian product XxY is the set of all ordered pairs (x,y) where x  X and y  Y In symbols XxY = {(x, y) | x  X and y  Y}  A binary relation R from a set X to a set Y is a subset of the Cartesian product XxY Example: X = {1, 2, 3} and Y = {a, b} R = {(1,a), (1,b), (2,b), (3,a)} is a relation between X and Y

Domain and range Given a relation R from X to Y,  The domain of R is the set Dom(R) = { x  X | (x, y)  R for some y  Y}  The range of R is the set Rng(R) = { y  Y | (x, y)  R for some x  X}  Example: if X = {1, 2, 3} and Y = {a, b} R = {(1,a), (1,b), (2,b)} Then: Dom(R)= {1, 2}, Rng(R) = (a, b}

Example of a relation  Let X = {1, 2, 3} and Y = {a, b, c, d}.  Define R = {(1,a), (1,d), (2,a), (2,b), (2,c)}  The relation can be pictured by a graph:

Properties of relations Let R be a relation on a set X i.e. R is a subset of the Cartesian product XxX  R is reflexive if (x,x)  R for every x  X  R is symmetric if for all x, y  X such that (x,y)  R then (y,x)  R  R is transitive if (x,y)  R and (y,z)  R imply (x,z)  R  R is antisymmetric if for all x,y  X such that x  y, if (x,y)  R then (y,x)  R

Order relations Let X be a set and R a relation on X R is a partial order on X if R is reflexive, antisymmetric and transitive. Let x,y  X  If (x,y) or (y,x) are in R, then x and y are comparable  If (x,y)  R and (y,x)  R then x and y are incomparable If every pair of elements in X are comparable, then R is a total order on X

Inverse of a relation Given a relation R from X to Y, its inverse R -1 is the relation from Y to X defined by R -1 = { (y,x) | (x,y)  R }  Example: if R = {(1,a), (1,d), (2,a), (2,b), (2,c)} then R -1 = {(a,1), (d,1), (a,2), (b,2), (c,2)}

2.5 Equivalence relations Let X be a set and R a relation on X  R is an equivalence relation on X  R is reflexive, symmetric and transitive. Example: Let X = {integers} and R be the relation on X defined by: xRy  x - y = 5. It is easy to show that R is an equivalence relation on the set of integers.

Partitions  A partition S on a set X is a family {A 1, A 2,…, A n } of subsets of X, such that A 1  A 2  A 3  …  A n = X A j  A k =  for every j, k with j  k, 1 < j, k < n.  Example: if X = {integers}, E = {even integers) and O = {odd integers}, then S = {E, O} is a partition of X.

Partitions and equivalence relations Theorem 2.5.1: Let S be a partition on a set X. Define a relation R on X by xRy if x, y are in the same set T for T  S. Then R is an equivalence relation on X. i.e. an equivalence relation on a set X corresponds to a partition of X and conversely.

Equivalence classes Let X be a set and let R be an equivalence relation on X. Let a  X.  Define [a] ={ x  X | xRa }  Let S = { [a] | a  X } Theorem 2.5.9: S is a partition on X.  The sets [a] are called equivalence classes of X induced by the relation R.  Given a, b  X, then [a] = [b] or [a]  [b] = 

Set of equivalence classes If R is an equivalence relation on a set X, define X/R = {[a] | a  X }. Theorem : If each equivalence class on a finite set X has k elements, then X/R has |X|/k elements, i.e. |X/R| = |X|/ k.

2.6 Matrices of relations  Let X, Y be sets and R a relation from X to Y  Write the matrix A = (a ij ) of the relation as follows: Rows of A = elements of X Columns of A = elements of Y Element a i,j = 0 if the element of X in row i and the element of Y in column j are not related Element a i,j = 1 if the element of X in row i and the element of Y in column j are related

The matrix of a relation (1) Example: Let X = {1, 2, 3}, Y = {a, b, c, d} Let R = {(1,a), (1,d), (2,a), (2,b), (2,c)} The matrix A of the relation R is A = abcd

The matrix of a relation (2)  If R is a relation from a set X to itself and A is the matrix of R then A is a square matrix.  Example: Let X = {a, b, c, d} and R = {(a,a), (b,b), (c,c), (d,d), b,c), (c,b)}. Then  A = abcd a1000 b0110 c0110 d0001

The matrix of a relation on a set X Let A be the square matrix of a relation R from X to itself. Let A 2 = the matrix product AA.  R is reflexive  All terms a ii in the main diagonal of A are 1.  R is symmetric  a ij = a ji for all i and j, i.e. R is a symmetric relation on X if A is a symmetric matrix  R is transitive  whenever c ij in C = A 2 is nonzero then entry a ij in A is also nonzero.

2.7 Relational databases  A binary relation R is a relation among two sets X and Y, already defined as R  X x Y.  An n-ary relation R is a relation among n sets X 1, X 2,…, X n, i.e. a subset of the Cartesian product, R  X 1 x X 2 x…x X n. Thus, R is a set of n-tuples (x 1, x 2,…, x n ) where x k  X k, 1 < k < n.

Databases  A database is a collection of records that are manipulated by a computer. They can be considered as n sets X 1 through X n, each of which contains a list of items with information.  Database management systems are programs that help access and manipulate information stored in databases.

Relational database model  Columns of an n-ary relation are called attributes  An attribute is a key if no two entries have the same value e.g. social security number  A query is a request for information from the database

Operators  The selection operator chooses n-tuples from a relation by giving conditions on the attributes  The projection operator chooses two or more columns and eliminates duplicates  The join operator manipulates two relations

2.8 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.

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  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.  Alternative definition: f : X  Y is one-to-one  for each pair of distinct elements x 1, x 2  X there exist two distinct elements y 1, y 2  Y such that f(x 1 ) = y 1 and f(x 2 ) = y 2. Examples: 1. The function f(x) = 2 x from the set of real numbers to itself is one-to-one 2. The function f : R  R defined by f(x) = x 2 is not one-to-one, since for every real number x, f(x) = f(-x).

Onto functions A function f : X  Y is onto  for each y  Y there exists at least one x  X with f(x) = y, i.e. Rng(f) = Y. Example: The function f(x) = e x 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.

Bijective functions A function f : X  Y is bijective  f is one-to-one and onto Examples:  1. A linear function f(x) = ax + b is a bijective function from the set of real numbers to itself  2. The function f(x) = x 3 is bijective from the set of real numbers to itself.

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) = x 2, 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.

Exponential and logarithmic functions  Let f(x) = 2 x 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(2 x ) = lg 2 x = x  Therefore, the exponential and logarithmic functions are inverses of each other.

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) = x 2 -1, f(x) = 3x + 5. Then f ◦ g(x) = f(g(x)) = f(3x + 5) = (3x + 5)  Composition of functions is associative: f ◦ (g ◦h) = (f ◦ g) ◦ h,  But, 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(x 1, x 2 )  X for every pair of elements x 1, x 2. 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: 1. 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).

String inverse Let X be any set, X* the set of all strings over X. If  = x 1 x 2 …x n  X*, let f(  ) =  -1 = x n x n-1 …x 2 x 1, the string written in reverse order. Then f :X*  X* is a function that defines a unary operator on X*. Observe that  -1 =  -1  =