Set and set operation

sets and set operations

标签（空格分隔）： mathematics

---

1.Discrete math(discrete structure leading up to sets)

A study of discrete structure used to represent discrete objects

and many discrete structures are built using sets.

• A collection of discrete objects.

2. sets

• (my own opinion:)a container for restoring a collection of elements which are unordered,not dupilicate.

• (formal definition:)set is a (unordered) collection of objects. These objects are sometimes called elements or members of the set.(Cantor's naive definition)

---

My rumination for "un-order " and "no duplicate element":

Main principle below:

Occam's Razor -> Entities should not be multiplied unnecessarily.

complexity : 1. un-ordered < ordered 2.single < duplicate

A question: Does higher complexity bring in new and efficient thing?

Obviously no any promotion in sets notation.sets only represents the discrete structure but no any need to depict how the structure of elements looks like.

---

3.relation between element and set

relation	description
e in set	\(e\in S\)
e not in set	\(e \notin S\)

Example :
A = [1,2,3,4]

\(1 \in A,5 \notin A\)

4.relation between sets

relation	description
subsets \(A\subseteq B\)	B consist of all the elements of A
proper subset \(A\subset B\)	B consists of all the elements of A and other elements
joint \(A\cap B \neq \emptyset\)	A and B are joint
disjoint\(A\cap B = \emptyset\)	A and B are disjoint

5.set operations

operations	description
union	\(A \cup B\)
intersection	\(A \cap B\)
complement	\(A^{'}\) : A' is the complement of A (i.e., all elements in S that are not in A).

5.1 the properties of set operations

commutative law \(A\cup B = B\cup A \quad \quad A \cap B = B \cap A\)
associative law \(A\cup (B \cup C) = (A \cup B) \cup C\)
distributive law \(A\cup (B\cap C) = (A\cup B)\cap(A\cup C)\)
-----------------\(A\cap (B \cup C) = (A\cap B)\cup(A\cap C)\)

De Morgan's Laws complement

\((A \cup B)^{'} = A^{'} \cap B^{'}\)

\((A \cap B)^{'} = A^{'} \cup B^{'}\)

6.Venn diagrams(sets visualization)

Venn diagrams usually are used to justify De morgan's laws.