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# Sets Definition of a Set: - PowerPoint PPT Presentation

Sets. Definition of a Set: NAME = {list of elements or description of elements} i.e. B = {1,2,3} or C = {x  Z + | -4 < x < 4} Axiom of Extension: A set is completely defined by its elements i.e. {a,b} = {b,a} = {a,b,a} = {a,a,a,b,b,b}. Subset. A B  xU, xAxB

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Presentation Transcript

Definition of a Set:

NAME = {list of elements or description of elements}

i.e. B = {1,2,3} or C = {xZ+ | -4 < x < 4}

Axiom of Extension:

A set is completely defined by its elements

i.e. {a,b} = {b,a} = {a,b,a} = {a,a,a,b,b,b}

AB  xU, xAxB

A is contained in B

B contains A

A B  x U, xA ^ xB

Relationship between membership and subset:

xU, xA  {x}  A

Definition of set equality: A = B  A B ^ B A

X={xZ | p Z, x = 2p}

Y={yZ | qZ, y = 2q-2}

A={xZ | i Z, x = 2i+1}

B={xZ | i Z, x = 3i+1}

C={xZ | i Z, x = 4i+1}

Set OperationsFormal Definitions and Venn Diagrams

Union:

Intersection:

Complement:

Difference:

Ordered n-tuple and the Cartesian Product

• Ordered n-tuple – takes order and multiplicity into account

• (x1,x2,x3,…,xn)

• n values

• not necessarily distinct

• in the order given

• (x1,x2,x3,…,xn) = (y1,y2,y3,…,yn)  iZ1in, xi=yi

• Cartesian Product

•  = alphabet = a finite set of symbols

• string over  =

empty (or null) string denoted as 

OR

ordered n-tuple of elements

• n = set of strings of length n

• * = set of all finite length strings

• Ø is a subset of every set.

• There is only one empty set.

• The union of any set with Ø is that set.

• The intersection of any set with its own complement is Ø.

• The intersection of any set with Ø is Ø.

• The Cartesian Product of any set with Ø is Ø.

• The complement of the universal set is Ø and the complement of the empty set is the universal set.

• Proper Subset

• Disjoint Set

A and B are disjoint

• A and B have no elements in common

• xU, xAxB ^ xBx A

AB = Ø  A and B are Disjoint Sets

• Power Set

P(A) = set of all subsets of A

• Inclusion

• Transitivity

• DeMorgan’s for Complement

• Distribution of union and intersection

Deriving new Propertiesusing rules and Venn diagrams

• A collection of nonempty sets {A1,A2,…,An} is a partition of the set A

if and only if

• A = A1A2…An

• A1,A2,…,An are mutually disjoint

Power set of A = P(A) = Set of all subsets of A

• Prove that

A,B {sets}, AB  P(A)  P(B)

• Prove that (where n(X) means the size of set X)

A {sets}, n(A) = k  n(P(A)) = 2k