The basic concept 1, the relationship
Relationship is a subset of the Cartesian product -> relationship is set as an ordered pair of elements
Property 2, relationship
Let R be a relation on a set A, then:
| nature |
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Relationship Matrix |
relation chart |
| Reflexive |
X belongs to any of A, XRX |
I_A in R |
Relationship matrix main diagonal elements are all 1 |
Diagram of all the points have loopback |
| Irreflexive |
Any X, Y belonging to A, XRY => X! = Y |
I_A∩R = sky |
Relationship matrix main diagonal elements are all 0 |
Graph all the points are not self-loop |
| symmetry |
Any X, Y belonging to A, XRY => YRX |
R=R^T |
All the main diagonal relationship matrix symmetric positions on the same element value |
There diagram A-> B must have B-> A |
| Antisymmetric |
Any X, Y belonging to A, XRY && YRX => X = Y |
R∩R ^ T belong to IA |
All the elements of the matrix relationship symmetrically about the main diagonal position are not simultaneously 1 |
If diagram A-> B must not B-> A |
| Transitive |
Any X, Y, Z belongs to A, XRY && YRZ => XRZ |
R=R·R |
slightly |
slightly |
| Symmetric and antisymmetric |
任意I,j,(I,j) = (j,i) = 0 |
| Asymmetric not antisymmetric |
任意<(I,j),(j,i)>取遍<0,0><0,1><1,0><1,1> |
| Does not reflexive anti-reflexive |
With a major diagonal 1 0 |
3 operation, relations
Computing a set of inherited (cross and makeup etc.) has its own operation (inverse operation R ^ -1, complex arithmetic R · T, the closure operation)
| Relations and operations (inherited from the collection) |
Matrix representation |
R∪S= MR ∨Ms |
Correspond directly to the position taken ∪ |
| Complex arithmetic relations |
Matrix representation |
R◦S= MR ÙMS |
A matrix multiplication, addition ∪ take place, the multiplication take ∩ |
| Inverse operation relations |
Matrix representation |
R^－1=(MR)T |
Relationship matrix inverse relationship is the relationship matrix transpose of the original relationship |
4, special relations
Let R be a relation on A
| Equivalence relation |
&& && symmetric reflexive transitive |
Determining a set of division |
Quotient set A / R |
A/R = { [a1]_R, [a2]_R,…} |
| Compatibility relation |
&& && contains symmetrical reflexive transitive |
Determine the coverage |
Maximal Compatible Classes |
Maximum complete polygon (n vertices FIG complete)
note:
A consistent relationship between the maximum compatibility class is not unique;
Vertices can be present in two compatible classes maximum inside (right) |
| Partial order |
&& && antisymmetric reflexive transitive |
|
Hasse |
Hasse is a chain: total ordering relation ; Each element of the A element has a minimum: well-ordered relationship ; - each of the finite set of totally ordered set is well-ordered Indeed bound upper bound lower bound on the lower boundary of what really is not a set of elements! |
| Coherent relationship |
Antireflexive && && contains transmittable antisymmetric |
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| A | = N, then the binary relation on A has 2 ^ (N * N) species
| Functional relationship |
X is an arbitrary X, Y such that there is always belong y f (x) == y |
Codomain X is called herein the domain of f, Y f is called, f is a subset of the range of Y ranf |
| Single-shot function |
f(x1) == f(x2) => x1 == x2 |
| X | = m, | Y | = n, if m> = n, then there is the number of X to Y is a single shot: C (m, n) * m! |
| Surjective function |
Ranf == Y, i.e. is equal range and codomain |
| X | = m, | Y | = n, if m <= n, then there is the number of X to Y is Surjective: C (n, 0) * (n ^ m) - C (n, 1) * ( (n-1) ^ m) + ... + C (n, n-1) * (1 ^ m) (repulsive force receiving) |
| Injective function |
Ranf a subset of Y |
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