Set Relations Functions 2009 Malestrom

7/23/2019 Set Relations Functions 2009 Malestrom

1/22

Sets, Relations and Functions 3

CHAPTER1

Sets, Relations and Functions

1.1 SYMBOLS AND NOTATIONS

Mathematics are full of symbols and notations. Symbols have their special meanings. So unless the

symbols are understood, no mathematical expression can be interpreted. So we begin with some symbols,

some of which mathematics has hired from logic. Students should always carry this collection as a tool

box in their memory.

: for all

: there exists

/ : there does not exist

! : there exists a unique

: belongs to

: does not belong to

: or (disjunction)

: and (conjunction)

: implies

: is implied by: implies and is implied by

iff : if and only if

i.e. : that is

viz. : namely

: such that

e.g. : for example

= : is equal to

: is not equal to

|| : is parallel to

: is perpendicular to.

1.2 SETS AND SET OPERATIONS

The notion of a set is basic in mathematics. We can express our ideas very precisely and concisely by

using this notion. We begin this notion here.


2/22

4 Topology

Definition: A setis a well-definedcollection of distinct objects.

The words well-defined and distinct are to be carefully noted here. These words were not originally

in the definition given by G. Cantor [1845 1918], who is known as the father of set theory. Bertrand

Russell pointed out some logical faults in the original definition and mended it as above. The wordwell-defined means unambiguously defined. Given any object, one should be able to determine whether

the object is within the collection or not. The collection of intelligent students of a school is not well

defined as opinions may differ in concluding who is intelligent and who is not.

The word distinct implies distinguishable with respect to some features or characteristics. Thus

the collection {1, 2, 3, 2} is not a set as the objects are not distinct.

Sets are usually denoted by capital letters or by putting the objects within the second brackets or

curly brackets. This is a universally accepted convention and therefore by no means the convention

should be disregarded. ThusX= {a, e, i, o, u} is a set but {1, 2, 3} is not a set. A set may be expressed

also by a property, e.g., {x;x is a prime number 10 }. This is the same as {2, 3, 5, 7}. If an element

belongs to a set, the fact of belonging is expressed symbolically by 2 1 2 3{ }, , . The fact of not belonging

similarly is expressed symbolically by 4 1 2 3{ }, , .Thus if we write x X , it shall mean the objectxis a member of the setX. Thus x X can be read

many ways as

(i) xbelongs toX

(ii) xis a point ofX

(iii) xis a member ofX

(iv) xis an element ofX

(v) xis an object ofX

(vi) xis contained inX.

The symbol y X can be accordingly interpreted.

For the sake of precision and consistency the notion of a set which has no object in its collection isaccepted in mathematics. This is known as the empty set and is usually denoted by the greek letter

(phi).

Thus the empty set is a set having no element.

Definition: A setA is called a subset of another setBif eitherAis the empty set or every element ofA

is also an element ofB.

Thus (i) The empty set is a subset of every set,

(ii) Every set is a subset of itself.

The fact thatAis a subset ofBis symbolically expressed as A B .

AsetBis called a supersetofAifA is a subset ofB. Note the notion of superset is just the opposite

of the notion of subset.

Observe, if A B and B C , then A C .

Two setsAandBare said to be equal, denoted byA=B, if A B and B A ,i.e., every element

ofAis contained inBand every element ofBis contained inA. If two setsAandBare not equal, we

express that symbolically by A B .

Thus the setsA= {a, b, c} andB= {c, a, b} are equal; but the sets P= {1, 2, 3, 4}, Q= {2, 3, 5, 6}

are not equal since 1 P but 1 Q.


3/22


Definition: To avoid logical contradictions in the long run all sets considered in any work are supposed

to be subsets of a large set. This large set is called the universal setof the system and kept fixed through

out the entire work.

Thus, when working with the setsA= {1, 2, 3},B= {2, 4, 5}, C= {1, 3, 5}, the set {1, 2, 3, 4, 5} orany superset of this may be taken as the universal set. But whichever is taken as the universal set must be

kept unchanged throughout the work.

Set Operations

There are three basic operations with sets; one is called the unary operationbecause it requires only one

set for its performance and two are binary operationsbecause they require two sets for their performance.

We define them as follows:

A x x Ac = { ; } . This set A

cis called the complementofAand the operation is sometimes

referred to as complementation.

A B x x A x B = { ; } . This set is called the union of the sets AandB and evidently

consists of those elements of the universal set, which belong to any one ofAandB.

A B x x A x B = { ; } . This set is called the intersectionof the setsAandB. Evidently

this consists of those elements of the universal set, which are common to bothAandB. Thus ifA= {1,

2, 3, 4},B = {2, 3, 5}, ={ }1 2 3 4 5 6, , , , , then

A Bc c= ={ } { }5 6 1 4 6, , , ,

A B ={ }1 2 3 4 5, , , ,

A B ={ }2 3, .

Note that c c= = , and ( )A Ac c = .Further note that A B B A A B B A = = , .

It is easy to see that

(i) A B B Ac c .

(ii) A A A = = , .(iii) A A A = = , .

(iv) A A A A A A = =, .

It is quite interesting and helpful to observe that the set operations through diagrams.

Because Venn used diagrams for the first time to visualize sets operations through diagrams such

diagrams are called Venn diagrams.

To visualize diagrammatically these set operations we take all points inside a rectangle as the universal

set and the region enclosed by a closed curve denotes a set. Thus we have

Ac

A B A B


4/22

6 Topology

The notion of set union and set intersections can be extended to finitely many or even infinitely

many sets.

Thus if A A An1 2, , , are nsets, then their union and intersection are denoted symbolically by

= ==i

i iA x x A n1

1 2{ ; for some i } , , ,

= ==i

i iA x x A i n1

1 2{ ; for each } , , ,

Therefore if A A A nn1 21 1 2 1 2= = ={ } { } { }, , , , , , , , then

==i

n

iA n1

1 2{ }, , ,

==i

n

iA1

1{ }

We now state some properties of set operations. LetA,B, C be three sets. Then

1. ( ) ( )A B C A B C = [Associative property of union]

2. ( ) ( )A B C A B C = [Associative property of intersection]

3. A B C A B A C = ( ) ( ) ( ) [Left distributive property]

4. A B C A B A C = ( ) ( ) ( ) [Left distributive property]

5. ( ) ( ) ( )A B C A C B C = [Right distributive property]

6. ( ) ( ) ( )A B C A C B C = [Right distributive property]

7. ( )A B A Bc c c = [De Morgans Law]

8. ( )A B A Bc c c = [De Morgans Law]

The De Morgans law is one of the finest properties of set operations and connects unions with

intersection. This property can be generalized as

( ) = A Ai c i c and ( ) = A Ai c i c

From the three basic operations we can define two more frequently used operation, called difference

and symmetric differenceas follows:

A B x A x B A B A B B A = = { ; } ( ) ( ) .

Observe that A B A Bc = and A B A B A B = .

Thus ifA= {1, 2, 3, 4},B= {2, 4, 5, 6},

then AB= {1, 3},B A= {5, 6}, A B ={ }1 3 5 6, , , .

Note from the definitions above it follows that

(i) A B B A , (ii) A B B A = .

Two sets are said to be disjointif their intersection is empty. ThusAandBare disjoint if A B =.Clearly the set {1, 2, 4} and {3, 5} are disjoint.

Thepower setof a set is the set of all subsets of the given set. The power set of the setXis usually

denoted by ( )X .

Thus ifX= {1, 2, 3}, then


5/22


=( ) { { } { } { } { } { } { } }X X, , , , , , , , , ,1 2 3 1 2 2 3 1 3 .

Note that X Y implies ( ) ( )X Y and conversely.

The Cartesian productof two setsAandB, written as A B is defined as a set whose elements areordered pairs (a, b) where a A b B .

Thus A B a b a A b B = {( ); }, , .

One can similarly define

A B C a b c a A b B c C = {( ); }, , , , .

As a convention we write A A A A A A A = =2 3, etc.

Note R2

is the set of order pairs of real numbers.

Finite and Infinite Sets

A set is calledfiniteif counting can exhaust its collection.

A set is infiniteif it is not finite.

Thus the set {a, e, i, o, u} is finite but the set of natural numbers is infinite.

The set of rational numbers and the set of integers are examples of infinite sets.

The number of elements in a finite set is called the cardinalityof the set and is usually denoted by

n(A) or card (A).

The following theorem, known as the Cardinality theorem, is of much practical importance.

Theorem 1.2.1: IfAandBare two finite sets, then

n A B n A n B n A B( ) ( ) ( ) ( ) = + .

Proof: Letxbe the number of elements common to bothAandB,ythe number of elements which are

inAbut not inBandzthe number of elements which are inBbut not inA.

Then n(A) =x+z. n(B) =y + z.

n A B x y z( ) = + + .

Evidently, n A B x y x z x( ) ( ) ( ) = + + +

= + n A n B n A B( ) ( ) ( ).

A B

y x z


6/22

8 Topology

Corollary 1: IfA,Band Care finite sets, then

n A B C n A n B n C n A B n B C ( ) ( ) ( ) ( ) ( ) ( ) = + +

n C A n A B C ( ) ( ) + .

Corollary 2: If { }Ai is a family of mfinite sets, then

n A n A n A A n A A Ai

ii

ii ji j

i ji j ki j k

i j k( ) ( ) ( ) ( ) = + = = =

=

1, = 2x 1 if x 1.

(iv) f x x( ) = | |1 defined on R

(v) f x x( ) = +[ ]1 defined on [2, 2].

Solution: (i) Sincey=f (x) = 2x+ 1 is linear inxandy, if represents a straight line.Two points on thisline can be found out by putting x= 0 and 1 and obtainingy= 1 and 3 respectively. So the points are

(0, 1) and (1, 3). Thus the graph is as follows:

1

0

2

3

4

5

6

1 2 3 4 x

f x( )


19/22


(ii) The expressionsy=x+ 1 andy= 1 xare linear and therefore represent parts of straight lines

respectively on ix 0 andx> 0. Fory=x+ 1, two points are (0, 1) and (1, 0) and fory= 1

x, two points are (1, 0) and (2, 1). Thus the graph will be as follows:

(iii) The expression y x= 2 represents a parabola passing through the origin and the expression

y = 2x1 represents a straight line. So the given function is made of two parts, one is a

segment of a parabola defined for x> 1, the other is a segment of a straight line for x 1.Thus the graph will be as follows:

3 2 1 1 2 3 X

1

2

3

Y

Graph of ( )f x

f x( )

n0

1

1

1

(0, 1)

y

x0


20/22

22 Topology

(iv) The functionf (x) = |x 1| can be rewritten as f x x x( ) if = 1 1, = 1 x ifx< 1. For x 1

the straight line segment will be determined by the points (0, 1) and (1, 0) and forx< 1, the

straight line segment will be determined by the points (0, 1) and (1, 2).

(v)

(vi) Here we observe that

f x x( ) if = < 1 2 1

=


21/22


At this moment the deep impact of this axiom cannot be understood but soon the application of this

axiom in proving some basic results will manifest its profundity.

1.6 COUNTABILITY

The notion of countability is an important tool in topology and analysis and needs to be grasped carefully.

Definition: A setXis said to be equivalent (or equipollent or equipotent) to the set Yif there exists a

bijective mapping fromXonto Y.

Since the inverse of a bijective mapping is also bijective, it follows readily that ifXis equivalent to

Y, then Yis equivalent toX.

This induces one to conclude that Xand Yare equivalent if there exists a bijective mapping from

one to the other. The fact thatXis equivalent to Yis expressed symbolically byX~ Y.

It follows readily that (i)X~X, (ii)X~ Yimplies Y~X, (iii)X~ Yand Y~ZimplyX~Z.

A rigorous definition of finite set is the following:

Definition: A set is said to be finite if it is equivalent to the set {1, 2, ..., n} for some natural number n.Analogously we define the following notions.

Definition: A set is said to be denumberableif it is equivalent to N. A set is called countableif it is

finite or denumberable.

The cardinality of Nis defined to be dor 0 (aleph null). Any set equivalent to Nis said to have the

same cardinality.

The cartesian product of two denumerable sets is also denumberable. Even the cartesian product of

a denumberable number of denumberable. The first result implies that the set Qof rational numbers is

denumberable and therefore has the cardinality dor 0 . The second result above induces the result that

the set of algebraic numbers is denumberable.

We shall see that a simple diagonal argument will prove that the set Rof real numbers is not

denumberable. The cardinal number of Ris defined to be cor 1 (aleph one). Any set equivalent to Ralso has the cardinality 1. Thus the set Iof irrational numbers has cardinarlity 1.

An interesting result of much use is the following:

Schroeder-Bernstein Theorem 1.6.1: IfXis equivalent to a subset of Yand Yis equivalent to a subset

ofX, thenXand Yare equivalent.

Proof: Let f be an injective mapping fromX to Y and gbe an injective mapping from Y to X. Let

x X . g x1( ), if it exiusts, is called the first ancestor ofx. For conventionxwill be called the zeroth

ancestor ofx. The element f g x 1 1{ ( )}, if it exists, will be called the second ancestor orx; the element

g1[ { ( )}]f g x

1 1 will be called the third ancestor ofx, In this way the succesive ancestors ofxcan be

defined. Clearly there are three posibilities: (1)xhas infinitely many ancestors, (2)xhas even number of

ancestors, (3)xhas odd number of ancestors. LetXidenote the set of elements ofXwhich have infinitely

many ancestors,Xethe set of elements ofXhaving even number of ancestors andXothe set of elementsofXhaving odd number of ancestors. Then X X X X X X Xi e o i e o= and , , are mutually disjoint.

We now define a function F X Y: as follows:

F xf x x X X

g x x X

i e

o

( )( ) if

( ) if =

,1

.


22/22

24 Topology

It is straight forward to prove now that Fis bijective. This proves the theorem.

With regard to cardinalities a natural question is whether there are (infinite) cardinals other than 0

and 1

. Cantor has provided the answer to this question.

Cantors Theorem:The power set of any set has cardinality greater than the cardinality of the set itself.

Proof:A mappingffrom a setXinto P (X) defined byf (x) = {x} proves that card ( ) card ( )X P X .We need to show that the inequality is strict. If possible suppose it is an equality that is,fis a bijective

mapping fromXonto P(X). We shall show then that a contradiction arises. Call a X a bad element if

a f a ( ). LetBdenote the set of all bad elements of X. Note B P X ( ). Sincefis surjective, there

exists b X such that f (b) =B. If b B , then bis a bad element, but then b f b B =( ) which is a

contradiction. Again if b B , then, b f b B =( ) , again a contradiction. So the assumption of theexistence of a bijective mapping is not tenable. This completes the proof.

Another relevant question in this regard is whether there is any cardinal between 0 and 1 .

Interestingly assumption of the absence of any such cardinal has been proved to be consistent with the

other axioms of set theory just as the negation has been proved to be consistent with the others as well.This assumption is known to be the Continuum Hypothesis.

Continuum Hypothesis: There is no setXsuch that < < 0 1card ( )X .

Documents

Set Relations Functions 2009 Malestrom