1. Discrete Mathematics
Relation

2. 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}

3. 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)}

4. 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

5. 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)

6. 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 |

7. 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

8. 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

9. 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}

10. 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:

11. 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

12. 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), …}

13. 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 =

14. 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

15. 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

16. 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

17. 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

18. 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.

19. 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)}

20. 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.

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

22. 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

23. 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

24. 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

25. 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.

26. 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.

27. 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.

28. 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

29. 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

30. 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.

31. 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.

32. 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).

33. 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.

34. 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.

35. 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.

36. 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 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.