Set Theory_Handout (1)

SET THEORY

SET THEORY

RICKY F. RULETE

Department of Mathematics and StatisticsUniversity of Southeastern Philippines

SET THEORY

SETS AND ELEMENTS, SUBSETS

A set may be viewed as any well-defined collection of objects,called the elements or members of the set.

sets are usually denoted by capital letters A,B, . . .

elements are denoted by lowercase letters a, b, . . .

membership in a set is denoted as follows:a S denotes that a belongs to a set Sa, b S denotes that a and b belong to a set S

Here is the symbol meaning is an element of. We use / tomean is not an element of.

SET THEORY


Specifying Sets

There are essentially two ways to specify a particular set.

Roster/Enumeration Method if possible, is to list its membersseparated by commas and contained in braces {}

Rule/Descriptive Method is to state those properties whichcharacterized the elements in the set

Illustration

Consider the following sets

A = {1, 3, 5, 7, 9} and B = {x | x is an even integer, x > 0}.

SET THEORY


Example (1.1)

1 The set A above can also be written asA = {x | x is an odd positive integer, x < 10}.

2 We cannot list all the elements of the above set B althoughfrequently we specify the set by B = {2, 4, 6, . . .}. Observethat 10 B, but 5 / B.

3 Let E = {x | x2 3x + 2 = 0}, F = {2, 1} andG = {1, 2, 2, 1}. Then E = F = G.

Remark

A set does not depend on the way in which its elements aredisplayed. A set remains the same if its elements are repeatedor rearranged.

SET THEORY


Subsets

Suppose every element in a set A is also an element of a set B,that is, suppose a A implies a B. Then A is called a subsetof B. We also say that A is contained in B or that B containsA. This relationship is written

A B or B A

Two sets are equal if they both have the same elements or,equivalently, if each is contained in the other. That is:

A = B if and only if A B and B AIf A is not a subset of B, that is, if at least one element of A

does not belong to B, we write A 6 B.

SET THEORY


Example (1.2)

Consider the sets:

A = {1, 3, 4, 7, 8, 9}, B = {1, 2, 3, 4, 5}, C = {1, 3}.

Then C A and C B since 1 and 3, the elements of C, arealso members of A and B. But B 6 A since some of theelements of B, e.g., 2 and 5, do not belong to A. Similarly,A 6 B.

SET THEORY


Property 1: It is a common practice in mathematics to put avertical line | or slanted line / through asymbol to indicate the opposite or negativemeaning of a symbol.

Property 2: The statement A B does not exclude thepossibility that A = B. In fact, for every set A wehave A A since, trivially, every element in Abelongs to A. However, if A B and A 6= B, thenwe say A is a proper subset of B (sometimeswritten A B).

Property 3: Suppose every element of a set A belongs to a setB and every element of B belongs to a set C.Then clearly every element of A also belongs to C.In other words, if A B and B C, then A C.

SET THEORY


Theorem (1.1)

Let A, B, C be any sets. Then:

(i) A A(ii) If A B and B A, then A = B

(iii) If A B and B C, then A C

SET THEORY


Special Symbols

N = the set of natural numbers : 0, 1, 2, . . .Z = the set of all integers: . . . ,2,1, 0, 1, 2, . . .Z+ = the set of positive integers: 1, 2, 3, . . .Z = the set of negative integers: 1,2,3, . . .Q = the set of rational numbersR = the set of real numbersR+ = the set of positive real numbersR = the set of negative real numbersC = the set of complex numbersObserve that N Z Q R C.

SET THEORY


Universal Set, Empty Set

All sets under investigation in any application of set theory areassumed to belong to some fixed large set called the universalset which we denote by U .Given a universal set U and a property P , there may not be anyelements of U which have property P . For example, thefollowing set has no elements:

S = {x | x Z+, x2 = 3}

Such a set with no elements is called empty set and is denotedby .

SET THEORY


Remark

1 There is only one empty set. That is, if S and T are bothempty, then S = T , since they have exactly the sameelements, namely, none.

2 The empty set is also regarded as a subset of every otherset.

Theorem (1.2)

For any set A, we have A U .

SET THEORY


Disjoint Sets

Two sets A and B are said to be disjoint if they have noelements in common. For example, suppose

A = {1, 2}, B = {4, 5, 6}, and C = {5, 6, 7, 8}

Then A and B are disjoint, and A and C are disjoint. But Band C are not disjoint since B and C have elements in common,e.g., 5 and 6.

Remark

If A and B are disjoint, then neither is a subset of the other(unless one is the empty set).

SET THEORY

VENN DIAGRAMS

Venn Diagram

Venn diagram is a pictorial representation of sets in whichsets are represented by enclosed areas in the plane.

The universal set U is represented by the interior of arectangle, and the other sets are represented by disks lyingwithin the rectangle.

U

AbB

Figure: Venn diagram of A B and b B.

SET THEORY

SET OPERATIONS

Union

The union of two sets A and B, denoted by A B, is the set ofall elements which belong to A or to B; that is

A B = {x | x A or x B}

Here or is used in the sense of and/or.

U U

A B A B

Figure: A B is shaded

SET THEORY

SET OPERATIONS

Intersection

The intersection of two sets A and B, denoted by A B, is theset of all elements which belong to both A and B; that is

A B = {x | x A and x B}

U U

A B A B

(a) (b)

Figure: A B is shaded

SET THEORY

SET OPERATIONS

Recall that sets A and B are said to be disjoint ornonintersecting if they have no elements in common or, usingthe definition of intersection, A B = . Suppose

S = A B and A B =

Then S is called the disjoint union of A and B.

SET THEORY

SET OPERATIONS

Example (1.3)

1 Let A = {1, 2, 3, 4}, B = {3, 4, 5, 6, 7}, C = {2, 3, 8, 9}.ThenA B = {1, 2, 3, 4, 5, 6, 7}, A C = {1, 2, 3, 4, 8, 9},B C = {2, 3, 4, 5, 6, 7, 8, 9}, A B = {3, 4},A C = {2, 3}, B C = {3}.

2 Let U be the set of students at a university, and let Mdenote the set of male students and let F denote the set offemale students. Then U is the disjoint union of M and F ;that is

U = M F and M F =

SET THEORY

SET OPERATIONS

Property 1: Every element x in A B belongs to both A andB; hence x belongs to A and x belongs to B. ThusA B is a subset of A and of B; namely

A B A and A B B

Property 2: An element x belongs to the union A B if xbelongs to A or x belongs to B; hence everyelement in A belongs to A B, and every elementin B belongs to A B. That is,

A A B and B A B

SET THEORY

SET OPERATIONS

Theorem (1.3)

For any sets A and B, we have

(i) A B A A B and (ii) A B B A B.

Theorem (1.4)

The following are equivalent: A B, A B = A,A B = B.

SET THEORY

SET OPERATIONS

Complement

The complement of a set A, denoted by Ac, is the set ofelements which belong to U but which do not belong to A; thatis

Ac = {x | x U , x / A}Some texts denote the complement of A by A or A.

A

U

Ac

Figure: Ac is shaded

SET THEORY

SET OPERATIONS

Relative Compliment

The relative complement of a set B with respect to a set X orsimply, the difference of A and B, denoted by A\B, is the set ofelements which belong to A but which do not belong to B; thatis

A\B = {x | x A, x / B}Some texts denote A\B by AB or A B.

U U

A B A B

(a) (b)

Figure: A\B is shaded

SET THEORY

SET OPERATIONS

Symmetric Difference

The symmetric difference of sets A and B, denoted by AB,consists of those elements which belong to Aor B but not both.That is

AB = (A B)\(A B) or AB = (A\B) (B\A)

U

A B

Figure: AB is shaded

SET THEORY

SET OPERATIONS

Example (1.4)

Suppose U = Z+ is the universal set. LetA = {1, 2, 3, 4}, B = {3, 4, 5, 6, 7}, C = {2, 3, 8, 9},E = {2, 4, 6, . . .}Then:Ac = {5, 6, 7, . . .}, Bc = {1, 2, 8, 9, 10, . . .}, Ec = {1, 3, 5, 7, . . .}AlsoA\B = {1, 2}, A\C = {1, 4}, B\C = {4, 5, 6, 7}, A\E = {1, 3},B\A = {5, 6, 7}, C\A = {8, 9}, C\B = {2, 8, 9},E\A = {6, 8, 10, 12}

SET THEORY

SET OPERATIONS

Example (1.4)

Furthermore:AB = (A\B) (B\A) = {1, 2, 5, 6, 7},B C = {2, 4, 5, 6, 7, 8, 9},A C = {1, 4, 8, 9},A E = {1, 3, 6, 8, 10, . . .}

SET THEORY

ALGEBRA OF SETS

Sets under the operations of union, intersection, andcomplement satisfy various laws (identities) which are listedbelow.

Idempotent laws: (1a) A A = A(1b) A A = A

Associative laws: (2a) (A B) C = A (B C)(2b) (A B) C = A (B C))

Commutative laws: (3a) A B = B A(3b) A B = B A

Distributive laws: (4a) A (B C) = (A B) (A C)(4b) A (B C) = (A B) (A C)

SET THEORY

ALGEBRA OF SETS

Identity laws: (5a) A = A(5b) A U = A(6a) A U = U(6b) A =

Involution law: (7) (Ac)c = A

Complement laws: (8a) A Ac = U(8b) A Ac = (9a) Uc = (9b) c = U

DeMorgans laws: (10a) (A B)c = Ac Bc(10b) (A B)c = Ac Bc

SET THEORY

ALGEBRA OF SETS

Remark

Each law listed above follows from an equivalent logical law.For example, the proof of DeMorgans Law 10(a):

(AB)c = {x | x / (A or B) } = {x | x / A and x / B} = AcBc

Here we use the equivalent(DeMorgans) logical law:

(p q) = p q

SET THEORY

FINITE SETS, COUNTING PRINCIPLE

A set is said to be finite if S is empty or if S contains exactly melements where m is a positive integer; otherwise S is infinite

Example (1.5)

1 The set A of the English alphabet and the set D of thedays of the week are finite sets. Specifically, A has 26elements and D has 7 elements.

2 Let E be the set of even positive integers, and let I be theunit interval, that is,

E = {2, 4, 6, . . .} and I = [0, 1] = {x | 0 x 1}

Then both E and I are infinite.

SET THEORY


The notation n(S) or |S| will denote the number of elements ina set S. Thus n(A) = 26, where A is the letters in the Englishalphabet, and n(D) = 7, where D is the days of the week. Alson() = 0 since the empty set has no elements.

Lemma (1.5)

Suppose S is the disjoint union of finite sets A and B. Then Sis finite and n(S) = n(A) + n(B).

SET THEORY


Corollary (1.6)

Let A and B be finite sets. Then n(A\B) = n(A) n(A B).

For example, suppose an art class A has 25 students and 10 ofthem are taking a biology class B. Then the number of studentsin class A which are not in class B is:

n(A\B) = n(A) n(A B) = 25 10 = 15

SET THEORY


Corollary (1.7)

Let A be a subset of finite universal set U . Thenn(Ac) = n(U) n(A).

For example, suppose a class U with 30 students has 18full-time students. Then there are 30 18 = 12 part-timestudent in the class U .

SET THEORY


Theorem (Inclusion-Exclusion Principle 1.8)

Suppose A and B are finite sets. Then A B and A B arefinite and

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

Corollary (1.9)

Suppose A, B, C are finite sets. Then A B C is finite and

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

SET THEORY


Example (1.6)

Suppose a list A contains the 30 students in a mathematicsclass, and a list B contains the 35 students in an English class,and suppose there are 20 names on both lists. Find the numberof students: (a) only on list A, (b) only on list B, (c) on list Aor B(or both), (d) on exactly one list.

SET THEORY


Solution:

(a) We seek n(A\B). By Corollary 1.6 we haven(A\B) = n(A) n(A B) = 30 20 = 10. Hence 10names are only on list A.

(b) We seek n(B\A). By Corollary 1.6 we haven(B\A) = n(B) n(A B) = 35 20 = 15. Hence 15names are only on list B.

(c) We seek n(A B). By inclusion-exclusion,n(A B) = n(A) + n(B) n(A B) = 30 + 35 20 = 45.

(d) By (a) and (b), 10 + 15 = 25 names are only on one list.

SET THEORY

CLASSES OF SETS, POWER SETS, PARTITIONS

Classes of Sets

Example (1.7)

Suppose S = {1, 2, 3, 4}.1 Let A be the class of subsets of S which contain exactly

three elements of S. Then

A = [{1, 2, 3}, {1, 2, 4}, {1, 3, 4}, {2, 3, 4}]

That is, the elements of A are the sets {1, 2, 3}, {1, 2, 4},{1, 3, 4}, and {2, 3, 4}.

SET THEORY


Classes of Sets

Example (1.7)

2 Let B be the class of subsets of S, each contains 2 and twoother elements of S. Then

B = [{1, 2, 3}, {1, 2, 4}, {2, 3, 4}]

The elements of B are the sets {1, 2, 3}, {1, 2, 4}, and{2, 3, 4}. Thus B is a subclass of A.

SET THEORY


Power Sets

The power set of set S is the class of all subsets of S, and willbe denoted by P (S). The number of elements in P (S) is 2raised to the power n(S). That is

n(P (S)) = 2n(S).

Example (1.8)

Suppose S = {1, 2, 3}. Then

P (S) = [, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, S].

SET THEORY


Partitions

Let S be a nonempty set. A partition of S is a subdivision of Sinto nonoverlapping, nonempty subsets. Precisely, a partition ofS is a collection {Ai} of nonempty subsets of S such that:

1 Each a in S belongs to one of the Ai.

2 The sets of {Ai} are mutually disjoint; that is, if

Aj 6= Ak then Aj Ak =

The subsets in a partition are called cells.

SET THEORY


Partitions

Example (1.9)

Consider the following collections of subsets ofS = {1, 2, . . . , 8, 9}:

1 [{1, 3, 5}, {2, 6}, {4, 8, 9}]2 [{1, 3, 5}, {2, 4, 6, 8}, {5, 7, 9}]3 [{1, 3, 5}, {2, 4, 6, 8}, {7, 9}]

Then (1) is not a partition of S since 7 in S does not belong toany of the subsets. Furthermore, (2) is not a partition of Ssince {1, 3, 5} and {5, 7, 9} are not disjoint. On the other hand,(3) is a partition of S.

SET THEORY


Generalized Set Operations

Consider first a finite number of sets, say, A1, A2, . . . , Am. Theunion and intersection of these sets are denoted and defined,respectively, by

A1 A2 . . . Am =mi=1

Ai = {x | x Ai for some Ai}

A1 A2 . . . Am =mi=1

Ai = {x | x Ai for every Ai}

SET THEORY



Now let A be any collection of sets. The union and theintersection of the sets in the collection A is denoted anddefined, respectively, by

(A | A A ) = {x | x Ai for some Ai A }(A | A A ) = {x | x Ai for every Ai A }

SET THEORY



Example (1.10)

Consider the setsA1 = {1, 2, 3, . . .}, A2 = {2, 3, 4, . . .},A3 = {3, 4, 5, . . .}, An = {n, n + 1, n + 2, . . .} Then

(Ak | k Z+) = Z+ and

(Ak | k Z+) =

SET THEORY



Theorem (1.10)

Let A be a collection of sets. Then:

1

[(A | A A )

]c=

(Ac | A A )

2

[(A | A A )

]c=

(Ac | A A )

SET THEORY


THANK YOU!!!

SETS AND ELEMENTS, SUBSETSVENN DIAGRAMSSET OPERATIONSALGEBRA OF SETSFINITE SETS, COUNTING PRINCIPLECLASSES OF SETS, POWER SETS, PARTITIONS

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Set Theory_Handout (1)