The intersection operation has two meanings:aggregateThe intersection operation of, that is, the intersection of two sets, which corresponds to theUnion operation, that is, of two setsUnion；It can also refer to the intersection operation of lattice, and the corresponding is latticeKnot operation。
Lattice is a kind of algebraic structure, which is based onPoset
Above, any element of E
The following two sets are constructed
{
And
}And
{
And
}As a subset of P, they are still partially ordered sets, and generally do not necessarily have minimum or maximum elements.If any element of P
，
AllMinimal element，
AllMaximum element, then P is calledgrid, and recorded as
。In case
Go, put
And its smallest element as a classBinary operation, called x and yhand over, recorded as
, symmetrically
The corresponding relationship with its largest element is regarded as the junction of x and y, which is recorded as
。They are the most basic operations on the lattice, and these two types of operations meet the following requirements:
1. Same:
2. Exchange law:
3. Combination law:
4. Absorption law:
among
And z are all arbitrary elements of E, so the lattice can also be regarded as an algebraic structure satisfying the above four laws
。
Although the theory of lattice was established relatively late, around the 1930s, it quickly developed in solving the problem of ordered sets, combinatorial problems and algebraic problems, and became an effective theoretical basis for relevant research. The development of lattice theory with matroid theory is an obvious example, which is different from the general algebraic structure with ordered characteristics,The order characteristics of elements in a lattice are not external, but internal, because their order relationship can be described by the internal operation of the lattice equivalently:
if and only if
perhaps
if and only if
This also reflects the symmetry of lattice intersection and junction operations.There are some important cases of lattices.For example,
Ge, here
Is a set family consisting of all subsets of E, and
, knot operation on it
Is the union and intersection of set x and set y
Is the intersection of set x and set y.As another example, if all positive divisors of the natural number n form elements of the set E, E
Ordered relation
If and only if x can divide y, then the partially ordered set
For any two sets A and B, the set consisting of all elements belonging to both A and B is called A and Bintersection, recorded as
。That is:
={
And
}。
Example
For example,
, then
Another example is:,
, then
If set
In other words, set A and set B have no common elements, so they are said to be disjoint.
For example:
that
That is, A and B do not intersect[2]。
Graphical representation
The operations between sets can be graphically represented by John Venn (British mathematician, 1834-1923).As shown in Figure 1, the rectangle on the plane represents the complete set
。Represented by a circle inside a rectangle
Any collection in.The shaded part in Figure 1 is
。
Figure 1 Intersection operation of sets
According to the definition of set intersection operation, intersection operation has the following properties:
(1) Idempotent law:
(2) Identical:
(3) Zero law:
(4) Binding law:
(5) Exchange law:
Similarly, the associative law can be generalized to the case of finite sets by induction[2]
