Discrete Mathematics Relation

Cartesian Product If A1, A2, …, Am are nonempty sets, then the Cartesian Product of these sets is the set of all ordered m-tuples (a1, a2, …, am), where ai  Ai, i = 1, 2, … m. Denoted A1  A2  …  Am = {(a1, a2, …, am) | ai  Ai, i = 1, 2, … m}

Cartesian Product Example If A = {1, 2, 3} and B = {a, b, c}, find A  B A  B = {(1,a), (1,b), (1,c), (2,a), (2,b), (2,c), (3,a), (3,b), (3,c)}

Subsets of the Cartesian Product Many of the results of operations on sets produce subsets of the Cartesian Product set Relational database Each column in a database table can be considered a set Each row is an m-tuple of the elements from each column or set No two rows should be alike

Using Matrices to Denote Cartesian Product For Cartesian Product of two sets, you can use a matrix to find the sets. Example: Assume A = {1, 2, 3} and B = {a, b, c}. The table below represents A × B. a b c 1 (1, a) (1, b) (1, c) 2 (2, a) (2, b) (2, c) 3 (3, a) (3, b) (3, c)

Cardinality of Cartesian Product The cardinality of the Cartesian Product equals the product of the cardinality of all of the sets: | A1  A2  …  Am | = | A1 |  | A2 |  …  | Am |

Subsets of the Cartesian Product Many of the results of operations on sets produce subsets of the Cartesian Product set Relational database Each column in a database table can be considered a set Each row is an m-tuple of the elements from each column or set No two rows should be alike

Introduction 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:

Example Paul Giblock R CSCI 2710 Danny Camper R CSCI 2710 A is a set of students and B is a set of courses A relation R may be defined as “register the course” Paul Giblock R CSCI 2710 Danny Camper R CSCI 2710

Relation on a Single Set Example A is the set of all courses A relation R may be defined as the course is a prerequisite CSCI 2150 R CSCI 3400 R = {(CSCI 2150, CSCI 3400), (CSCI 1710, CSCI 2910), (CSCI 2800, CSCI 2910), …}

Matrix of a Relation 1 if (ai, bj)  R 0 if (ai, bj)  R mij = We can represent a relation between two finite sets with a matrix MR = [mij], where 1 if (ai, bj)  R 0 if (ai, bj)  R mij =

Example Using the previous example where A = {1, 2, 3} and B = {a, b, c}. The matrix below represents the relation R = {(1, a), (1, c), (2, c), (3, a), (3, b)}. a b c 1 2 3

Digraph of a Relation Let R be a relation on A We can represent R using a diagram Each element of A is a circle called a vertex If ai is related to aj, then draw an arrow from the vertex ai to the vertex aj In degree means number of arrows coming into a vertex Out degree means number of arrows coming out of a vertex

Representing a Relation The following three representations depict the same relation on A = {1, 2, 3}. R = {(1, 1), (1, 3), (2, 3), (3, 2), (3, 3)} 1 2 3 1

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

Partial Order Relations Let X be a set and R a relation on X R is a partial order on X if R is reflexive, anti-symmetric and transitive.

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)}

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.

Equivalence classes Let X be a set and let R be an equivalence relation on X. Let a  X. Define [a] ={ xX | xRa }

Matrices of relations Let X, Y be sets and R a relation from X to Y Write the matrix A = (aij) of the relation as follows: Rows of A = elements of X Columns of A = elements of Y Element ai,j = 0 if the element of X in row i and the element of Y in column j are not related Element ai,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 = a b c d 1 2 3

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 = a b c d 1

The matrix of a relation on a set X Let A be the square matrix of a relation R from X to itself. Let A2 = the matrix product AA. R is reflexive  All terms aii in the main diagonal of A are 1. R is symmetric  aij = aji for all i and j, i.e. R is a symmetric relation on X if A is a symmetric matrix R is transitive  whenever cij in C = A2 is nonzero then entry aij in A is also nonzero.

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 X1, X2,…, Xn, i.e. a subset of the Cartesian product, R  X1 x X2 x…x Xn. Thus, R is a set of n-tuples (x1, x2,…, xn) where xk  Xk, 1 < k < n.

Databases A database is a collection of records that are manipulated by a computer. They can be considered as n sets X1 through Xn, 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

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’ E.g. Dom (f) = X = {a, b, c, d}, Range (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 Range (f)  Y. Therefore, no two pairs in f have the same first coordinate.

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 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  for each y  Y there exists at least one x  X with f (x) = y, i.e. Range (f) = Y. 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 Range (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) = x3 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) = 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.

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. Then f ◦ g (x) = f (g (x)) = f(3x + 5) = (3x + 5)2 - 1 Composition of functions is associative: f ◦ (g ◦h) = (f ◦ g) ◦ h, But, in general, it is not commutative: f ◦ g  g ◦ f.