Lecture 1

Lecture 1Learning ObjectivesTo use set notationsTo apply operations (union, intersection) on setsTo define de Morgans Laws for setsTo define relations on setsTo define set partitionsRMIT University, Taylor's University1Set TheorySet is a collection of objectsThe { } notation for sets

Example:X = { x | x is the alphabetical letter of English }

RMIT University, Taylor's UniversityThe element of SetThe objects are called the elements of a setWe use to denote the elements of a set.

x is an element of the set A

x is a member of A

x belongs to ARMIT University, Taylor's UniversityThe size of a set is called cardinality.The notation of cardinality is | |.Example:X = { x | x is the alphabetical letter of English }X = { a, b, c, , x, y, z }

| X | = 26

The element of Set

A singleton (a 1-element set)26 is the size or cardinality of XRMIT University, Taylor's UniversityThe Null Set / Empty Set

Examples: The set of numbers x such that x2 = -1.

The set of students who are doing Industrial Training and taking MATH 2111.RMIT University, Taylor's UniversityInfinite SetsThe set of all nonnegative integersN = {1, 2, 3, } (often used)N = {0, 1, 2, 3, } (in Maurer & Ralston)

The set of all integersZ = {, -2, -1, 0, 1, 2, }

The set of all rational numbersQ (quotients of integers or fractions)

The set of all real numbers, RCountably infiniteorCountableUncountably infinite or UncountableRMIT University, Taylor's UniversityNotations of Set

X is contained in Y

X is a subset of Y

Every element of X is also in Y.

RMIT University, Taylor's UniversityExample X = {1, 2, 3, 6, 7} Y = {4, 5, 6, 7, 8} Z = {2, 3, 6, 7}

RMIT University, Taylor's UniversityMore on Subsets

Note that the null set is regarded as a subset of every set, including itself.RMIT University, Taylor's UniversityExampleLet A = {a, b, c} = {b, c, a}. List all subsets of A.

(the no-element subset){a}, {b}, {c} (the 1-element subsets){a, b}, {a, c}, {b, c} (the 2-element subsets){a, b, c} (the 3-element subset)RMIT University, Taylor's UniversityThe 8-element Boolean AlgebraA{a, b}{a, c}{b, c}{a}{b}{c}RMIT University, Taylor's UniversityPower SetsThe set of all subsets of a given set X

Example: A = {a, b, c}

RMIT University, Taylor's UniversityTheorem

RMIT University, Taylor's UniversityOperations on SetsThe union of X and YThe set of all elements in X or Y

The intersection of X and YThe set of all elements in X and Y

RMIT University, Taylor's UniversityVenn Diagrams

XYXYRMIT University, Taylor's UniversityExample

a

b fcdeXYRMIT University, Taylor's UniversityComplement of Y relative to XX Y or X \ YThe set difference

XYRMIT University, Taylor's UniversityDisjoint SetsTwo sets are disjoint if they dont intersect

is called the disjoint union of X and Y

denoteRMIT University, Taylor's UniversityDisjoint Union of X and Y

XYRMIT University, Taylor's UniversityUniversal SetOften we have a universal set U consisting of all elements of interest.So every other set of interest is a subset of U.

XUThe complement of XRMIT University, Taylor's UniversityLemma: de Morgans Law for Sets

RMIT University, Taylor's UniversityExample U = {1, 2, , 10} X = {1, 2, 3, 4, 5} Y = {2, 4, 6, 8}Find:

RMIT University, Taylor's UniversityExamples of Some Other Sets

RMIT University, Taylor's UniversityCardinality of Set UnionsFor finite sets X and Y,

RMIT University, Taylor's UniversityRelations on SetsLet X, Y be sets. A relation between X and Y is a subset of the Cartesian product Let R be the relation from X to Y

So a relation is a set of ordered pairs of the form (x, y), where x X and y Y

RMIT University, Taylor's University25Cartesian coordinateRelations on Sets

Read as x rho y, to say that x is -related to yLet be a relation from x to y, andWe write,RMIT University, Taylor's UniversityExample 1

Any subset of R is a relation from X to Y.

and 61 moreLet R be the relation from X to Y.RMIT University, Taylor's UniversityExample 2:

Let

If we define a relation R from X to Y by

if y subtract x is a even number.

We obtain

The domain of R is

The range of R is

RMIT University, Taylor's UniversityRelation on SetsWhen X = Y, a relation between X and Y is called a relation on X

Any subset of X2 is a relation on X2 .

RMIT University, Taylor's UniversityExample 3

Let R be the relation on X

Define by if

Then,

y2357x2357XXRMIT University, Taylor's UniversityProperties of Relations: reflexiveLet be a relation on X. is reflexive if

Example 4:Let A = {1, 2, 3} and be a relation on A defined as

Therefore, is reflexive

y123x123AARMIT University, Taylor's UniversityLet be a relation on X.

is symmetric if

Example 5:Let A = {2, 3} and be a relation on A defined asx y if and only if x + y is odd integer.

Properties of Relations: symmetric

is symmetric RMIT University, Taylor's UniversityProperties of Relations: transitive is transitive if for all If and , and

Example 6:Let A = {1, 3, 4} and be a relation on A defined asx y if and only if

RMIT University, Taylor's UniversityEquivalence RelationsA relation on a set X is said to be an equivalence (relation) when it is reflexive, symmetric and transitive.Example 6 is a equivalence relation.

RMIT University, Taylor's UniversityCongruence modulo na is congruent to b modulo n when (a b) is an integer multiple of n

Usually we want a and b to be integers

for some integer tRMIT University, Taylor's UniversityExampleX = {1, 2, 3, 4, 5, 6, 7}. Define on X by x y if x y (mod 3). Write down as a set of ordered pairs.

= {(1,1), (1,4), (1,7), (2,2), (2,5), (3,3), (3,6), (4,1), (4,4), (4,7), (5,2), (5,5), (6,3), (6,6), (7,4), (7,7)}RMIT University, Taylor's UniversityCongruence modulo nIt can be shown that the relation a b (mod n) is always an equivalence relation on Z and its subsets

I.II.III.RMIT University, Taylor's UniversityApplications4 oclock + 12 hours = 1600 hours = 16 oclock = 4 oclockThis is because 4 16 (mod 12)

8 oclock + 12 hours = 2000 hours = 20 oclock = 8 oclockThis is because 8 20 (mod 12)

Coding theory is based on arithmetic modulo 2RMIT University, Taylor's UniversityPartitionsGiven a equivalence relation on a set X, we can partition X by grouping the related elements together.

A partition is a set of disjoint, nonempty subsets of a given set X whose union is X

Essentially, a partition divides X into subsets

RMIT University, Taylor's UniversityExample 6 (revisited)

X = {1, 2, 3, 4, 5, 6, 7}. Define on X by x y if x y (mod 3). Write down as a set of ordered pairs.

= {(1,1), (1,4), (1,7), (2,2), (2,5), (3,3), (3,6), (4,1), (4,4), (4,7), (5,2), (5,5), (6,3), (6,6), (7,4), (7,7)}Theorem:Equivalence classes of X given by the relation .

For every equivalence relation there is a corresponding partition, and vice versaRMIT University, Taylor's UniversityExample1 4

72 5 3 6 The partition corresponding to is often denoted by .

Here: = {{1,4,7}, {2,5}, {3,9}} RMIT University, Taylor's UniversityExampleConsider the following collections of subsets of S = {1, 2, 3, , 8, 9}:[{1, 3, 5}, {2, 6}, {4, 8, 9}][{1, 3, 5}, {2, 4, 6, 8}, {5, 7, 9}][{1, 3, 5}, {2, 4, 6, 8}, {7, 9}]Which of the above is a partition of S?RMIT University, Taylor's University42Only 3 is a partition of S.THE ENDRMIT University, Taylor's University