Kavita Hatwal Fall 20021 SET THEORY What is a set? A set is a collection of distinct objects. The objects in a set are called the elements or the members

Kavita Hatwal Fall 2002 1

SET THEORY

What is a set? A set is a collection of distinct objects. The objects in a set are called the elements or the members of the set. The name of the set is written in upper case and the elements of the set are written in lower case. If x is an element of a set A, we say that x belongs to or is a member of A, and is expressed symbolically as x A.If y is not a member of A, then this is symbolically denoted as y A

Let V be the set of all vowels. Then V is written asV= {a, e, i, o, u}

Name of the set

The curly bracketsDenoting the beginning

And end


The order in which elements appear do not matter, so{a,o,i,e, u}, {u, e, o, a, i}, {o, e, a, i,u} are all V

a {a} a is an element whereas {a} is a set whose element is a. {} makes all the difference in set notation. Page 242, #6-c, dPage 266, #6-c, dSets given by defining property.{x R | -2 < x < 5 }Is read as (from left to right) the set of all x such that x is a real number and also x is greater than –2 and less than 5. Can you list some of the members of this set?Page 242, #4-bPage 266, #4-bFinite and infinite sets.Sets whose elements can be listed are called finite sets, like H={seasons in a year}Which of the above sets are finite and which are infinite?


SUBSETS

If A and B are sets , A is called the subset of B, written as A B, if and only if, every element of A is also an element of B. Symbolically,A B , x, if x A then x B.Also A is contained in B or B contains A are ways of saying that A is a subset of B.Page 242, #6-g, hPage 266, #6-g, hEMPTY SETConsider the setX= Can you find any x which satisfies the above condition?A = {set of all animals}P={x A| x is a pink elephant}What could possibly be the elements of P?

}42,9:{ 2 xxx

A set with no elements is called an empty set denoted as . An empty Set is a subset of every set. A, where A is any set.


}{ because is the set with no elements {}, whereas { } is the set with one element, the empty set.

Subsets revisited

Let A and B be sets. A is a proper subset of B if, and only if, every element of A Is in B but there is at least one element of B that is not in A

For every set AA A, i.e. each set is its own subsetAnd A, i.e. the empty set is subset of every set

U is the universal set or universe of discourse. It is considered the all encompassing set. Every set is a subset of U.


Venn DiagramsWhat is a Venn Diagram? A Venn Diagram is a pictorial representation of sets. U is the universal set which is represented as a rectangle. Other sets are represented as circles.For example, if A and B are sets and A B, that is B contains A, then this situation is represented as

More on Venn Diagrams laterZ = set of integersQ = set of rational numbers.R= set of real numbersCan you show the subset relation between Z, Q and R using notation and Venn Diagram notation ?

A B


Set Equality

Venn Diagram

Page 243, #10-a, c Page 266, #10-a, c

Given two sets A and B, A equals B, written A = B, if, and only if, every element of A is in B and every element of B is in A.Symbolically,

ABandBABA

A=B

versus


Set Operations.Let A and B be subsets of the universal set U.1. The Union of A and B , denoted , is the set of all elements x in U such that x is either in A or in B. 2. The Intersection of A and B , denoted , is the set of all elements x in U such that x is in both A and B.3. The Difference of B minus A , denoted B-A, is the set of all elements x in U such that x is in B, but not in A.4. The complement of a set A, denoted as is the set of all elements x in U such that x is not in A. Symbolically

BA

BA

cA

}|{

} |{

} |{

} |{

AxUxA

BxandAxUxAB

BxandAxUxBA

BxorAxUxBA

c


Show the facts of previous slides using Venn Diagrams

Class ActivityDraw the rest on your own

Page 242, #7Page 266, #7

BA


c = U.

– Uc = .

– Complement

• A Ac = .

• A Ac = U.

– Distributivity

• A (B C) = (A B) (A C).

• A (B C) = (A B) (A C).

– Identity

• A U = A.

• A = A.

– Commutativity

• A B = B A.

• A B = B A.


Associativity

• (A B) C = A (B C).

• (A B) C = A (B C).

• Idempotent

– A A = A.

– A A = A.

• Universal Bounds

– A = .

– A U = U.

• Two sets A and B are called disjoint if they have no elements in common, i.e.

A B =

• Page 267, #6-a, 8

• Page 268, #19-a, 290 #4


Given a set A, the power set of A, denoted P (A), is the set of all subsets of A.

For all sets A and B, then1.2.

Page 268, #40Page 268, #26

Given 2 sets A and B, the Cartesian Product of A and B , denoted

, is the set of all orderedpairs(a,b), where a is in A and b is

in B.Given sets the

Cartesian Product of denoted by

is the set of all ordered n-tuples

where

Symbolically,

Page 238, example 5.1.10Page 265, example 5.1.15

BA

,,...,,A 21 nAA

,,...,,A 21 nAA

,...A 21 nAA ),...,,( 21 naaa

nn AaAaAa ,..., 2211

},...,

|),...,,{(...

},|),{(

2211

2121

nn

nn

AaAaAa

aaaAAA

BbAabaBA

)(P)(P, BABA

elements 2 has P(X)

then elementsn has Xset a if

, 0n integers, allFor

n


• Two sets A and B are disjoint if A B = .

• Sets A1, …, An are pairwise disjoint if Ai Aj = for all i, j {1, …, n}, with i j.

• A collection of sets {A1, …, An} is a partition of a set A if

– A1 … An = A, and

– A1, …, An are pairwise disjoint.


• Let – A0 = {n Z | n = 3k for some k Z}.

– A1 = {n Z | n = 3k + 1 for some k Z}.

– A2 = {n Z | n = 3k + 2 for some k Z}.

• Then {A0, A1, A2} is a partition of Z.


Set Identities


Boolean Algebras

• A Boolean algebra is a set S that

– includes two elements 0 and 1,

– has two binary operations + and ,

– has one unary operation ,which satisfy the following properties for all a, b, c in S:


Boolean Algebras

• Commutativity

– a + b = b + a.

– a b = b a.

• Associativity

– (a + b) + c = a + (b + c).

– (a b) c = a (b c).


Boolean Algebras

• Distributivity

– a (b + c) = (a b) + (a c)

– a + (b c) = (a + b) (a + c)

• Identity

– a + 0 = a.

– a 1 = a.


Boolean Algebras

• Complementation

– a + a = 1.

– a a = 0.


Examples: Boolean Algebras

• Let U be a nonempty universal set. Let 0 be and 1 be U. Let + be and be . Let be complementation. Then U is a Boolean algebra.

• Let U be a nonempty universal set. Let 0 be U and 1 be . Let + be and be . Let be complementation. Then U is a Boolean algebra.


Examples: Boolean Algebras

• Let S be the set of all statements. Let 0 be F and 1 be T. Let + be and be . Let be negation. Let = be . Then S is a Boolean algebra.

• Let S be the set of all statements. Let 0 be T and 1 be F. Let + be and be . Let be negation. Let = be . Then S is a Boolean algebra.


Derived Properties

• Theorem: Let S be a Boolean algebra and let a, b in S. Then– a a = a.– a + a = a.– a 0 = 0.– a + 1 = 1.– a b = a if and only if a + b = b.– a b = a + b if and only if a = b.


Derived Properties, continued

– a b = 0 and a + b = 1 if and only if a = b.– 0 = 1.

– 1 = 0.

– (a) = a.

– (a b) = a + b.– (a + b) = a b.


Example: Boolean Algebra

• Let S = {1, 2, 3, 5, 6, 10, 15, 30}.

• Define

– a + b = gcd(a, b).

– a b = lcm(a, b).

– a = 30/a.

– 1 is U, 30 is 0

• Verify the 10 basic properties.

• http://en.wikipedia.org/wiki/Greatest_common_divisor

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Kavita Hatwal Fall 20021 SET THEORY What is a set? A set is a collection of distinct objects. The objects in a set are called the elements or the members