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Survey of Mathematical Ideas Math 100 Chapter 2. John Rosson Thursday January 25, 2007. Basic Concepts of Set Theory. Symbols and Terminology Venn Diagrams and Subsets Set Operations and Cartesian Products Cardinal Numbers and Surveys Infinite Sets and Their Cardinalities.

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survey of mathematical ideas math 100 chapter 2

Survey of Mathematical IdeasMath 100Chapter 2

John Rosson

Thursday January 25, 2007

basic concepts of set theory
Basic Concepts of Set Theory
  • Symbols and Terminology
  • Venn Diagrams and Subsets
  • Set Operations and Cartesian Products
  • Cardinal Numbers and Surveys
  • Infinite Sets and Their Cardinalities
slide3

“a is an element of A”

“a is not an element of A”

Sets
  • A set is a collection of objects.
  • The objects in a set are called its elements or members.
  • If A is a set and a is an element of A, we show this in symbols as follows:
designating sets
Designating Sets
  • Word description:
  • Listing:
  • Set builder:
  • The empty set, designated Ø, is the set with no elements.

The set of positive whole numbers which are less than 20 and evenly divisible by 7.

set equality

This denoted, as one would expect

“set A equals set B”

“set A does not equal set B”

Set Equality

Set A is equal to set B if

  • every element of A is an element of B and
  • Every element of B is an element of A.
set equality1
Set Equality

Examples:

cardinality
Cardinality

The cardinal number or cardinality of a set is the number of element in a set.

In symbols, the cardinality of a set A is denoted

Equal sets have equal cardinality, but sets with equal cardinality are not always equal.

n(A)

cardinality1
Cardinality

Examples

Intuitively, the sets in the above example are finite. On the other hand, the sets of numbers N, Z, Q and R are all examples of infinite sets. Later, we precise definitions of the words “finite” and “infinite” mean.

subsets
Subsets

Set A is a subset of set B if every element of A is also an element of B

Denoted in symbols,

/

“set A is a subset of set B”

“set A is not a subset of set B”

If A and B are sets then A = B if A  B and B  A.

subsets1
Subsets

Set A is a propersubset of set B if A  B and A≠B.

This denoted in symbols,

“set A is a proper subset of set B”

“set A is not a proper subset of set B”

subsets2

/

Subsets

Examples. Let A be a set.

sets can be elements
Sets can be elements.

Any set can be an element of a set. If

then

power set

The power set of set A, denoted

is the set of all subsets of A. Thus

Power Set
power set1
Power Set

Example

In particular, the number of subsets of {1,2,3} is

power set2
Power Set

Theorem: The number of subsets of a finite set A is given by

and the number of proper subsets is given by