Create Time: 16th Jan 2024
Title: "[[sets-and-n-tuple]]"
status: DONE
Author:
- AllenYGY
tags:
- Sets
- Tuple
- DiscreteStructure
- Lec5
- NOTE
created: 2024-01-16T21:03
updated: 2024-04-08T19:39
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 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.
Power Sets
- The power sets of is which is the set of all the subsets of .
- The Cardinality of =
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