Equivalence, partial order, and full order

Equivalent:

SetRIs a collectionAA binary relationship. IfRThe following conditions are met:

1. Self-defense:
2. Symmetry:
3. Transmission:

It is calledRIs defined inA. Traditionally, the symbols of the equivalence relationship areRRewrite it to callback.

For example, set and defineALink onRAs follows:

It is calledXAndYModule
3. Same remainder, that isXDivide by the remainder of 3 andYDivided
The remainder of 3 is equal. The examples include 1R4, 2R5, and 3R6. Not difficult to verifyRIsA.

Not all binary relationships are equivalent. A simple inverse example is to compare which of the two numbers is larger:

• No self-inverse: no single number cannot be larger than itself ()
• No symmetry: IfM>N, You certainly cannot haveN>M

Partial OrderIs the binary relationship ≤ on the Set P, which is self-inverse, antisymmetric, and transmitted, that is, for all P's a, B, and c, there are:

• A ≤ a (self-inverse );
• If a ≤ B and B ≤ a, a = B (Anti-Symmetry );
• If a is less than or equal to B and B is less than or equal to c, a is less than or equal to c ).

A set with partial order is calledCollation set combination(Also calledPoset). TermsOrdered SetIt is also used for ordering Set combination, as long as the context does not involve the order of other types. In particular, full sequence sets can also be called "Ordered Sets", especially in areas where these structures are more commonly used than partial sequence sets.

Partial Order

Set A to A non-empty set, and P to A relation on A. If the relation P is self-inverse, called, and transmitted, P is the partial order relation on set. (# The difference between "add" and "equivalent" lies in the objection)

P is suitable for the following conditions:

(1) for any a in A, (a, a) in P;

(2) if (a, B) is P and (B, a) is P, a = B;

(3) if (a, B) is P, (B, c) is P, then (a, c) is P, P is A partial order relation on. A set with A partial order relationship is called A partial or semi-order set.

If P is A partial order relation on a, we use a ≤ B to represent (A, B) ε P.

The following example illustrates the partial order relationship:

1. The less than or equal relationship in the real number set is a partial order relationship.

2. Set S to a set, P (S) to a set composed of all subsets of S, and define P (S) if and only if A is A subset of B, that is, A is contained in B, P (S) becomes an ordinal set in this relationship.

3. If N is a positive integer set and m ≤ n is defined and only m can divide n, it is not difficult to verify that this is a partial order relationship. Note that it is different from the natural order relation on N.

Full Order

In set A, if there is an aRb or bRa for any a, B, and A, that is, every pair of elements in A satisfies the relationship R, then, the partial R on set A is in full or linear order.

The above is too scientific, and I cannot understand it. After a long time, I finally realized that it was such a thing.

A and B are examples in the book, which respectively represent partial order and full order. So what about ~~~~~ in the lower right corner ~~~~~ Amused ~~~ (* ^__ ^ *)

Let's see figure.

For normal traversal, there are two types of paths: 1234,1324, 2, and 3. They cannot determine who is before and who is before, and others can determine the order of the two. For example, 1 is always before 2 and 3. 2 and 3 are always before 4

What about 2 and 3? Unable to judge, so this is the partial order. At this time, because there is no order between 2 and 3, the whole is partially ordered. Then, we can see that B adds a direction between 2 and 3, from 2 to 3, so the route is only 1234,