2.1 Outline
- Denoting sets
- undefined term; set
- circular definition
- specifying sets
- description method
- roster method
- well-defined set
- terms
- members
- elements
- belong to
- contained in
- Sets of numbers
- natural numbers
- counting numbers
- integers
- rational numbers
- set-builder notation
- Equal and equivalent sets
- equal sets
- equivalent sets
- cardinality of a set
- cardinal number
- Universal and empty sets
- definitions
- notation for
- Venn diagrams
- complement
- general representation of a set
- Subsets and proper subsets
- definitions
- disjoint sets
- improper subset
- number of subsets
- general representation of two sets
- general representation of three sets
2.1 Essential Ideas
Denoting Sets
Sets are defined using the description or roster methods.
The objects in a set are called members or elements of the set.
The cardinality of a set is the number of elements in a set.
Two sets are equal if they contain the same number.
Two sets are equivalent if they have the same number of elements.
Sets of Numbers
Natural Numbers: {1, 2, 3, … }
Whole Numbers: {0, 1, 2, 3, … }
Integers: {…,, -2, -1, 0, 1, 2, … }
Rational Numbers: {a/b where a is an integer and b a nonzero integer}
Special Sets
The universal set contains all the elements under consideration in a given discussion.
The empty set contains no elements.
Venn Diagrams
One set divides the universe into 2 regions.
Two sets divide the universe into 4 regions.
Three sets divide the universe into 8 regions.
Complement
The complement of a set S is consists of everything that is not in S.
Subsets
A set A is a subset of a set B, if every element of A is also an element of B.
A set A is a proper subset of a set B, if every element of A is also an element of B and A and B are not equal sets.