sets-and-n-tuple

Sets

  • A set is a collection of objects
  • Sets are used to group objects together

Notation of Sets

  • {}

  • Expressions:

    • List all the members
    • Use predicates
    • Use suspension points

  • Properties：

    • No order
    • No duplicated elements

Universial Sets

  • A universal set[1] is a set that contains all the objects under consideration
  • Some Common universial sets
    • ℕ : the set of all natural numbers
    • ℤ : the set of integers
    • : the set of all the positive integers
    • ℚ: the set of all rational numbers
    • ℝ: the set of all the real numbers
    • ℂ: the set of all complex numbers

Venn Diagrams

  • A Venn diagram includes two basic shapes
    • A rectangle: indicates the universal set (all the objects under consideration)
    • Circles or other shapes: indicate normal sets.

Elements and sets

  • : x is in or is an element of S.
  • : x is not in or is not an element of S.

Subsets

  • Subsets
    • The set is a subsets of the set (denoted ) iff
  • iff
  • Proper Subsets
    • The set is a proper subsets of the set (denoted ) iff
  • Empty sets denoted

Cardinality

  • Cardinality is the number of distinct elements in a set.
    • The cardinality of a set is denoted as .
    • The Cardinality an be finite or infinite.
      • eg. ,

Power Sets

  • The power sets of is which is the set of all the subsets of .
  • The Cardinality of = [2]

Ordered n-tuple

  • The form or
    • Called ordered n-tuple
    • The elements in the tuple are ordered
    • E.g.,
      • is a 2-tuple is a 2-tuple they are different

Cartesian Product

  • Cartesian product of two sets and denoted()

  • if and , then

  1. 通用集,泛集↩︎
  2. use induction to proof↩︎
Table Of Contents
sets-and-n-tuple
Sets
Notation of Sets
Universial Sets
Venn Diagrams
Elements and sets
Subsets
Cardinality
Power Sets
Ordered n-tuple
Cartesian Product