Many different factors that binary relations

(that is, which has the following properties: each element of the set is equivalent to itself; if x equivalent to y, then y equivalent to x; if x equivalent to y, a y equivalent to z, then x equivalent to z ).

Then the set of all equivalence classes is called factor set and is denoted. The partition of a set into classes of equivalent elements is called its factorization.

Display from X into the set of equivalence classes is called factor mapping.

Examples

It is reasonable to use set factorization to obtain normed spaces from semi-normed spaces, spaces with an inner product from spaces with an almost inner product, etc. For this, the norm of a class is introduced, respectively, equal to the norm of an arbitrary element of it, and the scalar product of classes as the scalar product of arbitrary elements of classes. In turn, the equivalence relation is introduced as follows (for example, to form a normed quotient space): a subset of the original semi-normed space is introduced, consisting of elements with zero semi-norm (by the way, it is linear, that is, it is a subspace) and it is considered that two elements are equivalent if their difference belongs to this same subspace.

If for the factorization of a linear space some of its subspaces are introduced and it is assumed that if the difference of two elements of the original space belongs to this subspace, then these elements are equivalent, then the factor set is a linear space and is called a factor space.

Examples

§ 7. Equivalence relation. Factor set

A binary relation R on a set A is called an equivalence relation if R is reflexive, symmetric and transitive.

The relation of equality on the set of numbers has the indicated properties, therefore it is an equivalence relation.

The triangle similarity relation is obviously an equivalence relation.

The relation of non-strict inequality (≤ ) on the set of real numbers will not be an equivalence relation, because it is not symmetric: from 3 ≤ 5 it does not follow that 5 ≤ 3.

An equivalence class (coset) generated by an element a for a given equivalence relation R is the subset of those x A that are in relation R with a. The specified equivalence class is denoted by [a] R, therefore, we have:

[a] R = (x A: a, x R).

Consider an example. A similarity relation is introduced on the set of triangles. It is clear that all equilateral triangles fall into one coset, since each of them is similar, for example, to a triangle, all sides of which have unit length.

Theorem 1.6. Let R be an equivalence relation on a set A and [a] R be a coset, i.e. [a] R = (x A: a, x R), then:

1) for any a A : [a] R ≠ , in particular, a [a] R ;

2) different cosets do not intersect;

3) the union of all cosets coincides with the entire set A;

4) the set of different cosets form a partition of the set A.

Proof. 1) Due to the reflexivity of R, we obtain that for any a, a A, we have a, a R , therefore a [ a] R and [ a] R ≠ ;

2) suppose that [a] R ∩ [b] R ≠ , i.e. there is an element c from A and c [a] R ∩ [b] R . Then from (cRa)&(cRb), due to the symmetry of R, we obtain (aR c)&(cRb), and from the transitivity of R we have aRb.

For any х [а] R we have: (хRa)&(аRb) , then due to the transitivity of R we obtain хRb, i.e. x[b]R, so [a]R[b]R. Similarly, for any y, y [b] R , we have: (уRb)&(аRb) , and due to the symmetry of R we get that (уRb)&(bR а), then, due to the transitivity of R, we get that уR а , i.e. y[a]r and

so [b] R [a] R . From [a] R [b] R and [b] R [a] R we get [a] R = [b] R, i.e. if cosets intersect, then they coincide;

3) for any a, a A, as proved, we have a [ a] R , then it is obvious that the union of all cosets coincides with the set A.

Assertion 4) of Theorem 1.6 follows from 1)–3). The theorem has been proven. We can prove the following theorem.

Theorem 1.7. Different equivalence relations on a set A generate different partitions of A.

Theorem 1.8. Each partition of the set A generates an equivalence relation on the set A, and different partitions generate different equivalence relations.

Proof. Let a partition В= (B i ) of the set A be given. Let's define the relation R : a,b R if and only if there exists a B i such that a and b both belong to this B i . It is obvious that the introduced relation is reflexive, symmetric and transitive, hence R is an equivalence relation. It can be shown that if the partitions are different, then the equivalence relations generated by them are also different.

The set of all cosets of a set A with respect to a given equivalence relation R is called a quotient set and is denoted by A/R . The elements of the factor set are cosets. The coset class [ a ] R , as you know, consists of elements A that are in relation to each other R .

Consider an example of an equivalence relation on the set of integers Z = (…, -3, -2, -1, 0, 1, 2, 3, …).

Two integers a and b are called comparable (congruent) modulo m if m is a divisor of the number a-b, i.e. if we have:

a=b+km , k=…, -3, -2, -1, 0, 1, 2, 3, ….

In this case, write a≡ b(mod m) .

Theorem 1.9. For any numbers a , b , c and m>0 we have:

1) a ≡ a(mod m) ;

2) if a ≡ b(mod m), then b ≡ a(mod m);

3) if a ≡ b(mod m) and b ≡ c(mod m), then a ≡ c(mod m).

Proof. Statements 1) and 2) are obvious. Let us prove 3). Let a=b+k 1 m , b=c+k 2 m , then a=c+(k 1 +k 2 )m , i.e. a ≡ c(mod m) . The theorem has been proven.

Thus, the relation of comparability modulo m is an equivalence relation and divides the set of integers into non-overlapping classes of numbers.

Let us construct an infinitely unwinding spiral, which in Fig. 1.13 is depicted with a solid line, and an infinitely twisting spiral, depicted with a dashed line. Let a non-negative integer m be given. We place all integers (elements from the set Z ) at the intersection points of these spirals with m rays, as shown in Fig. 1.13.

For the relation of comparability modulo m (in particular, for m = 8) the equivalence class is the numbers lying on the ray. Obviously, each number falls into one and only one class. It can be obtained that for m= 8 we have:


[ 0] ={…, -8, 0, 8, 16, …};
[ 1] ={…, -7, 1, 9, 17, …};
[ 2] ={…, -6, 2, 10, 18, …};

[ 7] ={…, -9, -1, 7, 15, …}.

The set factor of a set Z with respect to comparison modulo m is denoted as Z/m or as Z m . For the case under consideration, m =8

we obtain that Z/8 = Z8 = ( , , , …, ) .

Theorem 1.10. For any integers a, b, a * , b * , k and m :

1) if a ≡ b(mod m), then ka ≡ kb(mod m);

2) if a ≡ b(mod m) and a* ≡ b* (mod m), then:

a) a + a * ≡ b + b * (mod m); b) aa * ≡ bb* (mod m).

We present the proof for case 2b). Let a ≡ b(mod m) and a * ≡ b * (mod m) , then a=b+sm and a * =b * +tm for some integers s and t . Multiplying,

we get: aa* =bb* + btm+ b* sm+ stm2 =bb* +(bt+ b* s+ stm)m. Hence,

aa* ≡ bb* (mod m).

Thus, modulo comparisons can be added and multiplied term by term, i.e. operate in exactly the same way as with equalities. For example,

∼ (\displaystyle \sim ). Then the set of all equivalence classes is called factor set and is denoted. The partition of a set into classes of equivalent elements is called its factorization.

Display from X (\displaystyle X) into the set of equivalence classes X / ∼ (\displaystyle X/\!\sim ) called factor mapping. Due to the properties of the equivalence relation, the partition into sets is unique. This means that classes containing ∀ x , y ∈ X (\displaystyle \forall x,\;y\in X) either do not intersect or coincide completely. For any element x ∈ X (\displaystyle x\in X) some class is uniquely defined from X / ∼ (\displaystyle X/\!\sim ), in other words, there exists a surjective mapping from X (\displaystyle X) in X / ∼ (\displaystyle X/\!\sim ). The class containing x (\displaystyle x), sometimes denoted [ x ] (\displaystyle [x]).

If the set is provided with a structure, then often the mapping X → X / ∼ (\displaystyle X\to X/\!\sim ) can be used to supply a factor set X / ∼ (\displaystyle X/\!\sim ) the same structure, such as topology. In this case, the set X / ∼ (\displaystyle X/\!\sim ) with the induced structure is called quotient space.

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Subtitles

Factor space by subspace

Often the equivalence relation is introduced as follows. Let be X (\displaystyle X)- linear space , and L (\displaystyle L) is some linear subspace. Then two elements x , y ∈ X (\displaystyle x,\;y\in X) such that x − y ∈ L (\displaystyle x-y\in L), are called equivalent. This is denoted x ∼ L y (\displaystyle x\,(\overset (L)(\sim ))\,y). The space obtained as a result of factorization is called quotient space by subspace L (\displaystyle L). If a X (\displaystyle X) expands into a direct sum X = L ⊕ M (\displaystyle X=L\oplus M), then there is an isomorphism from M (\displaystyle M) in X / ∼ L (\displaystyle X/\,(\overset (L)(\sim ))). If a X (\displaystyle X) is a finite-dimensional space, then the quotient space X / ∼ L (\displaystyle X/\,(\overset (L)(\sim ))) is also finite-dimensional and dim ⁡ X / ∼ L = dim ⁡ X − dim ⁡ L (\displaystyle \dim X/\,(\overset (L)(\sim ))=\dim X-\dim L).

Examples

. We can consider the factor set X / ∼ (\displaystyle X/\!\sim ). Function f (\displaystyle f) sets a natural one-to-one correspondence between X / ∼ (\displaystyle X/\!\sim ) and Y (\displaystyle Y).

Let G=(p 0 =e, p 1 , …, p r ) be some permutation group defined on the set X = (1, 2, …, n) with identity e=p 0 by the identical permutation. We define the relation x~y by setting x~y, which is equivalent to saying that there exists a p belonging to G(p(x)=y). The introduced relation is an equivalence relation, that is, it satisfies three axioms:

1) x~x;
2) x~y→y~x;
3) x~y&y~z→x~z;

Let A be an arbitrary set.
Definition: A binary relation δ=A*A is an equivalence relation (denoted a ~ b) if they satisfy the following axioms:
∀ a, b, c ∈ A
1) a ~ a - reflexivity;
2) a ~ b ⇒ b ~ a - commutativity;
3) a ~ b & b ~ c → a ~ c - transitivity

denoted by a ~ b, σ(a,b), (a,b) ∈ σ, a σ b

Definition: A partition of a set A is a family of pairwise disjoint subsets from A, in union (in sum) giving all A.
А= ∪А i , А i ∩А j = ∅, ∀i ≠ j.

The subsets A i are called cosets of the partition.

Theorem: each equivalence relation defined on A corresponds to some partition of the set A. Every partition of the set A corresponds to some equivalence relation on the set A.

Briefly, there is a one-to-one correspondence between the classes of all equivalence relations defined on the set A and the class of all partitions of the set A.

Proof: let σ be an equivalence relation on a set A. Let a ∈ A.

Let's build a set: К a =(x ∈ A,: x~a ) – all elements equivalent to a. The set (notation) is called the equivalence class with respect to the equivalence σ. Note that if b belongs to K a , then b~a. Let us show that a~b⇔K a =K b . Indeed, let a~b. Take an arbitrary element c belongs to K a . Then c~a, a~b, c~b, c belongs to K b and therefore K b belongs to K a . The fact that K a belongs to K b is shown similarly. Therefore, K b =K a .
Let now K b =K a . Then a belongs to K a = K b , a belongs to K b , a~b. Which is what needed to be shown.

If 2 classes K a and K b have a common element c, then K a = K b . Indeed, if c belongs to K a and K b , then b~c, c~a, b~a => K a = K b .

Therefore, different equivalence classes either do not intersect or intersect and then coincide. Every element c of A belongs to only one equivalence class K c. Therefore, the system of non-overlapping equivalence classes at the intersection gives the entire set A. And therefore this system is a partition of the set A into equivalence classes.

Converse: Let A = sum over or A i be a partition of A. Let's introduce the relation a~b on A, as a~b ⇔ a,b belong to the same partition class. This relation satisfies the following axioms:

1) a ~ a (are in the same class);
2) a ~ b → b ~ a;
3) a ~ b & b ~ c → a ~ c, i.e. the introduced relation ~ is an equivalence relation.

Comment:
1) the partition of the set A into one-element subsets and the partition of A, consisting only of the set A, is called a trivial (improper) partition.

2) The division of A into one-element subsets corresponds to the equivalence relation, which is equality.

3) Partition A, consisting of one class A, corresponds to an equivalence relation containing A x A.

4) a σ b → [a] σ = [b] σ — any equivalence relation defined on some set divides this set into pairwise disjoint classes called equivalence classes.

Definition: The set of equivalence classes of the set A is called the factor set A/σ of the set A by the equivalence σ.

Definition: A mapping p:A→A/σ such that p(A)=[a] σ is called a canonical (natural) mapping.

Any equivalence relation defined on a set divides this set into pairwise disjoint classes, called equivalence classes.

Let R be a binary relation on a set X. The relation R is called reflective , if (x, x) О R for all x О X; symmetrical – if (x, y) О R implies (y, x) О R; the transitive number 23 corresponds to the variant 24 if (x, y) Î R and (y, z) Î R imply (x, z) Î R.

Example 1

We will say that x н X has in common with element y н X if the set
x З y is not empty. The relation to have in common will be reflexive and symmetrical, but not transitive.

Equivalence relation on X is called a reflexive, transitive, and symmetric relation. It is easy to see that R Н X ´ X will be an equivalence relation if and only if the inclusions take place:

Id X Í R (reflexivity),

R -1 Í R (symmetry),

R ° R Í R (transitivity).

In fact, these three conditions are equivalent to the following:

Id X Í R, R -1 = R, R ° R = R.

splitting set X is a set A of pairwise disjoint subsets a н X such that UA = X. With each partition of A, we can associate an equivalence relation ~ on X by setting x ~ y if x and y are elements of some a н A.

To each equivalence relation ~ on X there corresponds a partition A, whose elements are subsets, each of which consists of those in the relation ~. These subsets are called equivalence classes . This partition A is called the factor set of the set X with respect to ~ and is denoted: X/~.

Let us define the relation ~ on the set w of natural numbers by setting x ~ y if the remainders after dividing x and y by 3 are equal. Then w/~ consists of three equivalence classes corresponding to remainders 0, 1, and 2.

Order relation

A binary relation R on a set X is called antisymmetric , if from x R y and y R x follows: x = y. A binary relation R on a set X is called order relation , if it is reflexive, antisymmetric and transitive. It is easy to see that this is equivalent to the following conditions:

1) Id X Í R (reflexivity),

2) R Ç R -1 (antisymmetry),

3) R ° R Í R (transitivity).

An ordered pair (X, R) consisting of a set X and an order relation R on X is called partially ordered set .

Example 1

Let X = (0, 1, 2, 3), R = ((0, 0), (0, 1), (0, 2), (0, 3), (1, 1), (1, 2 ), (1, 3), (2, 2), (3, 3)).

Since R satisfies conditions 1–3, then (X, R) is a partially ordered set. For elements x = 2, y = 3, neither x R y nor y R x is true. Such elements are called incomparable . Usually the order relation is denoted by £. In the example above, 0 £ 1 and 2 £ 2, but it is not true that 2 £ 3.

Example 2

Let be< – бинарное отношение строгого неравенства на множестве w натуральных чисел, рассмотренное в разд. 1.2. Тогда объединение отношений = и < является отношением порядка £ на w и превращает w в частично упорядоченное множество.

Elements x, y О X of a partially ordered set (X, £) are called comparable , if x £ y or y £ x.

The partially ordered set (X, £) is called linearly ordered or chain if any two of its elements are comparable. The set in Example 2 will be linearly ordered, but the set in Example 1 will not.

A subset A Í X of a partially ordered set (X, £) is called bounded from above , if there exists an element x н X such that a £ x for all a н A. An element x н X is called greatest in X if y £ x for all y О X. An element x О X is called maximal if there are no elements y О X different from x for which x £ y. In example 1, elements 2 and 3 will be the maximum, but not the largest. The bottom constraint subsets, least and minimum elements. In example 1, element 0 would be both the least and the minimum. In example 2, 0 also has these properties, but (w, t) has neither the greatest nor the maximum element.

Many different factors that binary relations

Examples

Examples

see also

See what "Factorset" is in other dictionaries:

§ 7. Equivalence relation. Factor set

Encyclopedic YouTube

Subtitles

Factor space by subspace

Examples