If setAAny element of is a collectionBElement of (anya∈Abea∈B）, then setAIs called a setBOfsubset, marked as A ⊆ B orB⊇A, read as "setAcontainGather onB”Or setBInclude CollectionA”。
If setAyesBA subset of, andA≠B, i.eBAt least one element in does not belong toA, thenAnamelyBOfProper subset, can be recorded as:A⊂ B。[4]
Symbolic language: Ruo ∀a∈A. Averageyesa∈B, and
X ∈ B makes x ∉ A, thenA⊊B。
Figure 1
As shown in Figure 1, set A is the proper subset of set B.[2]
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1、 According to the definition of subset, we know thatA⊆A。in other words,Any set is a subset of itself。
2、 For the empty set ∅, we specify ∅ ⊆A，I.eempty setIs anyaggregateSubset of。
Note: If A=∅, ∅ ⊆ A is still valid.
Proof: Given any set A, it is necessary to prove that ∅ is a subset of A.This requires that all ∅ elements are A elements;However, ∅ has no element.For experienced mathematicians, it is obvious to infer that "∅ has no elements, so all elements of ∅ are elements of A"; but for beginners, there is some trouble. Because ∅ has no elements, how to make "these elements" become elements of other sets? Another way of thinking will help.
In order to prove that ∅ is not a subset of A, you must find an element that belongs to ∅ but does not belong to A.Because ∅ has no element, this is impossible.Therefore ∅ must be a subset of A.
This proposition shows that for any set S, SPower setThe order by inclusion is a bounded lattice. If it is combined with the above proposition, it is aBoolean algebra。
5、 : For any twoaggregateA and B, all of the following expressions are equivalent:
A ⊆ B
A ∩ B =A
A ∪ B = B
A − B=A (when A ∨ B=∅);A − B=C 𝖠 (A 𝖠 B) (when A 𝖠 B ≠∅)
B′ ⊆ A′
This proposition explains: expressing "A ⊆ B" and other usesUnion, intersection andComplementThe expression of is equivalent, that is, the inclusion relation is redundant in the axiom system.
6、 AssumptionsNon empty setAWithnElements, there are:
The number of subsets of A is 2n。
The number of proper subsets of A is 2n-1。
The number of non empty subsets of A is 2n-1
The number of non empty proper subsets of A is 2n-2。[3]
