Generally, we use uppercase subtitles to represent sets, such as A and B, and lowercase letters to represent elements, such as A and B. ..
Of course, a set itself can also be an element of another set.
If all the elements in set A are elements in set B, then set A is called a subset of b with the symbol A? B or B? A, pronounced as a contained in b or b contained in a. A∈A has a∈B, what about A? B.
According to the definition of subset, we know that a? A. In other words, any set is a subset of itself.
For empty sets? We specify a, that is, an empty set is a subset of any set.
Proper subset:
If the set A is a subset of B, and A≠B, that is, at least one element in B does not belong to A, then A is the proper subset of B, which can be written as: A? B.
Extended data:
What if? A, b and c are sets, then:
Reflexivity:? Answer? A, antisymmetry:? AB and? BA if and only if A=? B. transitivity: if? AB and? What about BC? Communication. This proposition states: For any set? S, the power set of S is a bounded lattice containing order. Combining the above propositions, it is a Boolean algebra.
What if? A, b and c are sets? S, then:
There are minimum elements and maximum elements:? AS( A is given by proposition 2). Existence and operation:? AA∪B if? AC and? What about BC? A∪BC has intersection operation:? A∩BA if? CA and? CB？ The proposition of CA∩B shows that the expression "AB" is equivalent to other expressions that use union, intersection and complement, that is, the inclusion relation is redundant in the axiomatic system.
An empty set is a subset of any set.
Proof: Given any set A, what do you want to prove? Is it an a? A subset of. It takes all? Is the element a? Elements of; But? No elements.
For experienced mathematicians, inference "? No elements. So? Obviously, all the elements of are elements of A; But for beginners, there are some troubles. It would be helpful to think differently, to prove it? Is not a subset of, you must find the element to which it belongs. , but it doesn't belong to A. Because? There is no element, so this is impossible. So what? It must be.
This proposition shows that inclusion is a partial order relationship.
References:
Baidu encyclopedia-subset
References:
Baidu Encyclopedia-proper subset