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Table 1 set identities Identity A⋃ø=A, A⋂U=A A⋃U=U, A⋂ø=ø A⋃A=A, A⋂A=A (A)=A A⋃B=B⋃A, A⋂B=B⋂A A⋃(B⋃C)=(A⋃B) ⋃C A⋂(B⋂C)=(A⋂B) ⋂C A⋂(B⋃C)=(A⋂B) ⋃(A⋂C) A⋃(B⋂C)=(A⋃B) ⋂(A⋃C)
Theorem 1
For any set S, 1. ø⊆S 2. S⊆S
When we wish to emphasize that a set A is a subset of the set B but that A≠B, we write A⊂B and say that A is a proper subset of B. One way to show that two sets have the same elements is to show that each set is a subset of the other. In other words, we can show that if A and B are sets with A⊆B and B⊆A, then A=B.That is: ∀x(x∈A →x∈B) ,∀x(x∈B →x∈A)
Name
A⋃B=A⋂B ,A⋂B=A⋃B
Let A and B be sets. The difference of A and B, denoted by A-B, is the set containing those elements that are in A but not in B. The difference of A and B is also called the complement of B with respect to A. An element x belongs to the difference of A and B if and only if x∈A and x∉B. So: A-B={x|x∈A∧x∉B}.
Definition
Let S be a set, If there are exactly n distinct elements in S where n is a nonnegative integer, we say that S is a finite set and that n is the cardinality of S. The cardinality of S is denoted by |S|. A set is said to be infinite if it is not finite.
P({0,1,2})={ ø, {0},{1},{2},{0,1},{0,2},{2,1},{0,1,2}}
What is the power set of the empty set?What is the power set of the set {ø}?
Definition
The ordered n-tuple (a1,a2,...,an) is the ordered collection that has a1 as its first element, a2 as its second element, ..., and an as its nth element. We say that two ordered n-tuples are equal if and only if each corresponding pair of their elements is equal.
The Power Set
Given a set S, the power set of is the set of all subsets of the set S. The power set of S is denoted by P(S).
Example
What is the power set of set {0,1,2}?
Definition
Two set are called disjoint if their intersection is the empty set. |A⋃B|=|A|+|B|-|A⋂B|:
principle of inclusion-exclusion.
（包含排斥原理）
Definition
Example
Show that the Cartesian product B ⅹ A is not equal to the Cartesian product A ⅹ B , where A and B are as: A={1,2}, B={a,b,c}
Definition
The Cartesian product of the sets A1,A2,...,An, denoted by A1ⅹA2 ⅹ... ⅹAn, is the set of ordered n-tuples (a1,a2,...,an), where ai belongs to Ai for i=1,2,...,n, In other words A1ⅹA2 ⅹ... ⅹAn={(a1,a2,...,an)|ai∈Ai,for i=1,2,...,n}.
Definition
The set A is said to be a subset of B if and only if every element of A is also an element of B. We use the notation A⊆B to indicate that A is a subset of the set B. We see that A⊆B if and only if the quantification: ∀x(x∈A →x∈B)
Definition1
A set is an unordered collection of objects.
Definition 2
The objects in a set are also called the elements, or members, of the set. A set is said to contain its elements.
Draw a Venn diagram that represents V, the set of vowels in the English alphabet
U aei ou
Definition
A set has no elements is called the empty set, or null set, and is denoted by ø. The empty set can also be denoted by { }. You should pay attention to ø and {ø}.
Set builder notation
The set of all real numbers:
R={x | x is a real number}
Venn diagrams 文氏图
In Venn diagrams the universal set U, which contains all the objects under consideration, is represented by a rectangle. Inside this rectangle, circles or other geometrical figures are used to represent sets. Sometimes points are used to represent the particular elements of the set.
Example
What is the Cartesian product AⅹB ⅹC, where A={0,1}, B={1,2}, and C={0,1,2}?
2.2 Set Operations
Definition 1 Let A and B be sets. The union of the sets A and B, denotes by A⋃B, is the set that contains those elements that are either in A or in B, or in both. A⋃B={x|x∈A∨x∈B}.
How to describe a set?
All vowels in the English alphabet :
V={ a, e, i, o, u}
Definition 3
Two set are equal if and only if they have the same elements. Note that the order in which the elements of a set are listed does not matter.
Example
What is the Cartesian product of A={1,2} and B={a,b,c}? A ⅹ B= {(1,a),(1,b),(1,c),(2,a),(2,b),(2,c)}.
The Cartesian products A ⅹ B and B ⅹ A are not equal, unless A=ø or B=ø or unless A=B.
Name Identity laws 恒等律 Domination laws 支配律 Idempotent laws 幂等律 Complementation laws 补集律
Commutative laws交换律 Associative laws结合律
Distributive laws分配律
Identity
