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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 ProductThe 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 xX and yY In symbols XxY = {(x, y) | xX and yY}
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 rangeGiven a relation R from X to Y, The domain of R is the set
Dom(R) = { xX | (x, y) R for some yY}
The range of R is the set Rng(R) = { yY | (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 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 We can represent a relation between two
finite sets with a matrix MR = [mij], where
1 if (ai, bj) R0 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 1 0 1
2 0 0 1
3 1 1 0
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 RelationThe following three representations depict the same relation on A = {1, 2, 3}.
R = {(1, 1), (1, 3), (2, 3), (3, 2), (3, 3)}
1 0 1
0 0 1
0 1 1
1
2
3
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 xX 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,yX such that
xy, if (x,y) R then (y,x) R
Partial Order RelationsLet 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 relationGiven 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] ={ xX | 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 1 0 0 1
2 1 1 1 0
3 0 0 0 0
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
a 1 0 0 0
b 0 1 1 0
c 0 1 1 0
d 0 0 0 1
The matrix of a relation on a set XLet 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.