For example, the intersection of sets {1, 2,3} and {2,3,4} is {2,3}. The number 9 does not belong to the intersection of the prime set {2,3,5,7, 1 1} and the odd set {1, 3,5,7,9,1}.
If the intersection of two sets A and B is empty, that is, they have no common elements, then they do not intersect.
More generally, intersection operations can be performed on multiple collections at the same time. For example, the intersection of sets a, b, c and d is a ∩ b ∩ c ∩ d = a ∩ (b ∩ (c ∩ d)). The intersection operation satisfies the associative law, that is, a ∩ (b ∩ c) = (a ∩ b) ∩ c c.
The most abstract concept is the intersection of arbitrary non-empty sets. If m is a non-empty set and its elements are also sets, then X belongs to the intersection of m if and only if it belongs to a for any element of m.
In other branches of set theory and mathematics, the set is not a set composed of all the elements of these sets, but does not contain other elements.
Get ready.
If A and B are sets, the union of A or B is a set containing all elements of A and B, but no other elements. The union of a and b is usually written as "A ∪B".
Formally: X is an element of A ∪B, if and only if X is an element of A, or X is an element of B.
For example, the union of sets {1, 2,3} and {2,3,4} is {1, 2,3,4}. The number 9 does not belong to the union of the prime set {2, 3, 5, 7, 1 1, …} and the even set {2, 4, 6, 8, 10, …}, because 9 is neither prime nor even.
More generally, the union of multiple sets can be defined as follows: for example, the union of A, B and C contains all elements of A, all elements of B and all elements of C, but no other elements.
Formally: X is an element of A ∪B ∪C if and only if X belongs to A or X belongs to B or X belongs to C.
Algebraic properties: Binary union (union of two sets) is a combinatorial operation, that is, A ∨( B∪C)=(A∪B)∪C, in fact, A ∪B ∪C is also equal to these two sets, so parentheses can be omitted when only union operation is performed.
Similarly, the union operation satisfies the exchange rate, that is, the order of the sets is arbitrary.
Empty set is the unit element of union operation. That is, {} ∪A = A, for any set A. An empty set can be regarded as the union of zero sets.
Joint operation combines intersection operation and complement set operation, and integrates any power into Boolean algebra. For example, union and intersection satisfy the distribution law, and these three operations satisfy De Morgan's law. If the union operation is replaced by symmetric difference operation, the corresponding Boolean ring can be obtained.
Infinite union: The most common concept is the union of any set. If M is a set, then X is an element of M's union, if and only if there is an element A of M and X is an element A ... that is: x \ in \ big cup \ mathbf \ iff \ exists an {\ in} \ mathbf, X \ in a.
For example, A ∪ B ∪ C is the union of sets {A, b, C}. At the same time, if m is an empty set, the union of m is also an empty set. The concept of finite union can be extended to infinite union.
There are many ways to express the above concepts: set theory scientists simply write \bigcup \mathbf, while most people will write \bigcup_{A\in\mathbf} A like this. The latter writing can be extended to \bigcup_{i\in I} A_, which means the union of the set {Ai: i in I}. Here I is a set, and Ai is a set to which I belong. When the exponent set I is a set of natural numbers, the above expression is similar to summation: \ bigcup _ {I = 1} {\ infty} a _.
Similarly, it can also be written as "a1∪ A2 ∪ A3 ∪ ..." (This is an example of the combination of countable sets, which is very common in mathematical analysis; See σ-algebra). Finally, it should be noted that when the symbol "∨" is placed before other symbols, but not between them, it should be written larger.