Symbolic language: if? A∈A, there are all a∈B, what about A? B.
Chinese name
subset
Foreign name
subset
application area
Mathematical science
Application category
gather
express
A∈A, there are all a∈B, what about A? B
quick
navigate by water/air
nature
definition
If any element of set A is an element of set B (any a∈A is a∈B), then set A is called a subset of set B, and is denoted as A? B or B? A, read as "set A is contained in set B" or set B contains set A ".
Namely:? A∈A has a∈B, what about A? B. [1]
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.
Symbolic language: if? A∈A, there are all a∈B, x∈B makes X? A, then a? B.
As shown in the overview, set A is the proper subset of set B [2].
nature
First of all, according to the definition of subset, we know that a? A. In other words, any set is a subset of itself.
Second, for the empty set? We specify a, that is, an empty set is a subset of any set.
Description: If A=? Then a is still valid.
Proof: Given any set A, what do you want to prove? Is a subset of a. This requires giving all. The element of is the element 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. Because? Without any elements, how can "these elements" become elements of other collections? Another way of thinking will help.
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.
3. If A, B and C are sets, then:
Reflexivity: A=A
Anti-symmetry: If and only if,
Transitivity: If and, then
This proposition shows that inclusion is a partial order relationship.
Fourth,
This proposition shows that the power set of any set s, s is an ordered bounded lattice, and when combined with the above proposition, it is a Boolean algebra.
V: For any two sets A and B, all the following statements are equivalent:
Answer? B
A ∩ B =A
A ∪ B = B
Answer? B=A (when A∩B=? ) ; Answer? B=C？ (A∩B) (when A∩B≦? )
b′？ one
This proposition states: the expression "A? B "is equivalent to other expressions that use union, intersection and complement, that is, the inclusion relation is redundant in the axiomatic system.
6. Assuming that the nonempty set A contains n elements, there are:
The number of subsets of a is 2n.
The number of proper subset of A is 2n- 1.
The number of nonempty subsets of A is 2n- 1.
The number of nonempty proper subset of A is 2n-2. [3]
reference data
[1] Suzhou Education Press. Compulsory mathematics 1. Suzhou: Suzhou Education Press, 20 12.
[2] People's Education Press. Mathematics required 1. Beijing: People's Education Press, 20 12.
[3] Qu Yixian. Three-year simulation of five-year college entrance examination. 20 14: Beijing Capital Normal University Press