stayset theoryAt least oneelementOfaggregate, calledNon empty set, called non empty set for short.In other words, exceptempty setOther sets are non empty sets.[1]
If a set is not an empty set, it is called a non empty set, for example
Both are non empty collections.[2]
Related concepts
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aggregate
aggregateIt is one of the basic concepts of mathematics.It is the foundation of modern mathematics.Some objects with certain attributes are regarded as a whole, forming aaggregate, set short namecollection。[1]
For example: (1) All students in Grade One of Yuying Primary School can form a collection;(2) All natural numbers also form a set;(3) The four great inventions of ancient China form a collection.
Sets are generally represented by capital letters A, B, C.
Elements of a collection
The objects that make up a collection are calledCollectivelyelement。
For example: (1) The collection of first grade students in Yuying Primary School is composed of each student as its element;(2) 1, 2, 3,... are elements of the set of natural numbers;(3) Compass, gunpowder, printing and paper making are all elements of the four great inventions of ancient China.
Generally, lower case letters a, b, c,... are used to represent the elements of a set.
Relationship between elements and sets
If a is an element of set A, it is said that element abelong toSet A, denoted as a ∈ A.The symbol "∈" means belongs to, read as "a belongs to A", or read as "A contains a";If a is not an element of set A, it is said that a does not belong to A and is recorded as a
A. Symbol“
”RepresentDoes not belong to, read as "a does not belong to A", or read as "A does not contain a".
Sets usually have the following three representations:
(1)Enumerative methodList the elements in the set one by one and write them in {}.
Example 1Positive of 6DivisorThe set A of A={1, 2, 3, 6}
(2)Descriptive methodDescribe the common characteristics of the elements in the set and write them in {}.
Example 2The set C of all odd numbers can be expressed as: C={x | x=2n+1, and n is an integer}.
(3)Venn diagrammethodThe method of using a closed curve to circle all the elements in the set to represent the set is calledvenn diagrams 。
Example 3A set of four elements, a, b, c, d, can be represented by a Venn diagram.As shown on the left in Figure 1.
Figure 1
Example 4The collection of colors that make up the pattern of the Chinese national flag can be represented by a Venn diagram, as shown on the right of Figure 1.
For example, a set consisting of a: {a} is aSingle element set, for example, a set composed of 0: {0} is a single element set.
subset
If any element of set A is an element of set B, set A is called set Bsubset。record as
or
, read as "A included in B".Or "B contains A".
For example, A={1, 2, 5}, B={1, 2, 3, 5, 7}, because every element of set A is an element of set B.So A is a subset of B, that is
。
For example, A={triangle}, B={isosceles triangle}. Then
。
be careful:For a set A, each element belongs to itself, so it has
In other words, any set is a subset of itself.In addition, an empty set is a subset of any set.
Proper subset
If set A is a subset of set B, and at least one element of set B does not belong to set A, then set A is called set BProper subset, recorded as
or
。Read as A really contains B or B really contains A.
For example: A={1,2,5}, B={1,2,3,5,7}, set A is a subset of set B.At least one element (element 3) in set B does not belong to set A, so set A is the proper subset of set B, that is
or
。
Equal set
For two sets A and B, if
, similarly
Let's say that these two sets are equal, recorded as A=B, and read as "A equals B".
For example;A={0, 1, 2}, B={2, 0, 1}, since every element in set A is an element of set B, it means that set A is a subset of set B, that is
。Similarly,
, so A=B.
intersection
A set consisting of all the common elements of set A and set B is called theintersection。record as
, read as A and hand in B.useVenn diagramIt is shown in Figure 2 (shaded part):
Figure 2
For example, A={1, 2, 3, 5, 7}, B={2, 5, 6, 10}, then
For example, A={1,3}, B={4,5,6}
The Venn diagram is shown in Figure 3.
Figure 3
Union
A set consisting of all elements of set A and set B is called set A and set BUnion, recorded as
, read as A and B.The Venn diagram is shown as follows: (shaded part)[1]
