In the sacred world, mathematics is called

proper subset

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If set A is a subset of set B, and at least one element in set B does not belong to a, then set A is called the proper subset of set B. If A is contained in B and A is not equal to B, then set A is called the proper subset of set B. ..

Chinese name

proper subset

Foreign name

proper subset

Another name

True tolerance

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Answer? B

Applied discipline

mathematics

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1 definition

subset

proper subset

for instance

2 related propositions

definition

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subset

Generally speaking, for two sets A and B, if any element in set A is an element in set B, we say that these two sets have an inclusion relationship, and call set A a subset of set B ... and call it a? B (or b? A), pronounced "A is contained in B" (or "B contains A"). [ 1]

That is to say, for sets a and b,? X∈A has x∈B, so what about A? B. [1] It is known that any set A is a subset of itself and an empty set is a subset of any set. [2] [3]

proper subset

If you set one? B, there is an element x∈B, and the element X does not belong to the set A. We say that the set A and the set B have a true inclusion relationship, and the set A is the proper subset of the set B, which is denoted as A? B (or b? A), pronounced "A really contains B" (or "B really contains A").

That is to say, for sets a and b,? X∈A has x∈B, and? X∈B and x A, then a? B an empty set is a proper subset of any non-empty set.

Non-empty proper subset: If a is set? B, and set A≦? , set A is the nonempty proper subset of set B [2]

The difference between proper subset and subset:

A subset is that all elements in one set are elements in another set, which may be equal to another set;

Proper subset means that all elements in one set are elements in another set, but there is no equivalence. [ 1]

for instance

The collection of all Asian countries is the collection of all countries on earth in proper subset; The set of all natural numbers is the proper subset of the set of all integers (that is, n? z)； { 1, 3} ? { 1, 2, 3, 4},{ 1, 2, 3} ? { 1, 2, 3, 4}; {? }。 But you can't say {1, 2,3}? { 1, 2, 3}。 [2]

Let the complete set I be {1, 2,3}, then its subsets can be {1}, {2}, {3}, {1, 2}, {1, 3}, {2,3}. ; And its proper subset can only be {1}, {2}, {3}, {1, 2}, {1, 3}, {2,3},? . Its non-empty proper subset can only be {1}, {2}, {3}, {1, 2}, {1, 3}, {2,3}. [ 1]

Related proposition

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Proposition 1: If set A has n elements, then

example

The number of subsets of set A is 2n, with 2n- 1 proper subset and 2n-2 non-empty proper subset. [ 1]

Proof: Let the number of elements be 1, 2, ... n, and each subset corresponds to a binary number of length n (the I-th bit of the specified number is 1, which means that element I is in the set, and 0 means that element I is not in the set. If the complete set U={e 1, e2, e3, e4, e5}, then {e 1, e2, e3, e4, e5}? 1 1 1 1 1，{e2，e3，e4}？ 0 1 1 10，{e4}？ 000 10)。 That is to say, its subset is 00...0(n zeros) ~11...1(n1). It is easy to know that a * * * has 2n numbers, so it corresponds to 2n subsets. Delete 1 1... 1 (that is, the original set A) with 2n- 1 proper subset, and then remove 00...0 (that is, the empty set) with 2n-2 non-empty proper subset. [4]

Proposition 2: An empty set is a subset of any set.

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. 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. [4]

Proposition 3: If A, B and C are sets, then:

Reflexivity: a? A, anti-symmetry: a? B and b? A, if and only if A= B, transitivity: if a? B and b? C is a? C. This proposition shows that the power set of any set s, s is a bounded lattice containing order, and when combined with the above proposition, it is a Boolean algebra.

Proposition 4: If A, B and C are subsets of the set S, then: [4]

There is a minimum element and a maximum element: a? S( A is given by proposition 2). Existence and operation: a? A∪B if a? C and b? C is A∪B? C has an intersection operation: A∩B? A if c? A and c? B is c? A ∩ B. 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. [3]

Proposition 5: For any two sets A and B, the following statements are equivalent: A? B A∩ B= A A∪ B= B A？ B = B′？ [2] One

reference data

1. Qu Yixian, etc. Three-year simulated mathematics, which must be taken in the five-year college entrance examination. Beijing: Capital Normal University Press, 20 14.

2. People's Education Press. Compulsory mathematics 1. Beijing: People's Education Press, 20 14.

3. Lu Baoxian. Elementary mathematics. Beijing: Peking University Publishing House, 20 16.

4. Flex Klein. Looking at elementary mathematics from a high angle. Shanghai Fudan University Press 20 10

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Natural science, science and technology, science

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