Ordered relationships. Strict order relation

The last example introduces the concept of alphabetical order.

Alphabet is a tuple of pairwise distinct characters called letters of the alphabet. An example is the alphabet of any European language, as well as the alphabet of 10 Arabic numerals. In a computer, the keyboard and some aids determine the alphabet of valid characters.

Word in the alphabetA - tuple of alphabet characters A. The word is written with alphabetic characters in a row, from left to right, without spaces A natural number is a word in the digital alphabet A formula is not always a word due to the non-linear arrangement of characters the presence of superscript (exponents) and subscript (indices of variables, bases of logarithms) symbols, fractional bar, signs radicals, etc.; however, by some conventions, it can be written into a string, which is used, for example, in computer programming (for example, the exponentiation sign is written as 2 multiplication signs in a row: 5**3 means the third power of the number 5.

Lexico-graphic (alphabetic) ordering - for various words in the alphabet with ordered

characters set ordering: if

possible presentation , at which either

(subword can be empty), or - empty subword

In this definition - a prefix (initial subword) that is the same for both words - or the first in a row on the left are different

characters, or - the last character in the word - tail

subwords.

Thus, the alphabetical ordering of words is determined by the first character that distinguishes them from the left (for example, the word KONUS precedes the word COSINUS, since they first differ in the third letter, and H precedes C in the Russian alphabet). It is also considered that the space character precedes any character of the alphabet - for the case when one of the words is a prefix of the other (for example, KOH and CONE)

Exercise. Check that alphabetical ordering natural numbers, having the same number of digits in decimal notation, coincides with their ordering in magnitude.

Let A - partially ordered set. The element is called maximum V A, if there is no element for which A< b. Element A called greatest V A, if for any other than A item completed b<а-

are defined symmetrically minimum and least elements. The concepts of the largest and maximum (respectively, the smallest and minimum) elements are different - see. example in Fig.14. The set in Fig. 14a has the largest element R, it is also the maximum, there are two minimum elements: s and t there is no smallest. In Fig. 14b, on the contrary, the set having two maximum elements / and j , there is no greatest, minimum, it is the smallest - one: T.

In general, if a set has a largest (respectively, smallest) element, then only one (there may be none).

There can be several maximum and minimum elements (may not be at all - in an infinite set; in the final case, there must be).

Let's look at two more examples. - relation on the set N:

"Y divides X", or "X is the divisor of the number Y"(For example,

) is reflexive and transitive. Consider it on a finite set of divisors of the number 30.

The relation is a relation of partial order (non-strict)

and is represented by the following matrix of order 8, containing 31 characters

The corresponding scheme with 8 vertices must contain 31 bundles. . However, it will be more convenient for viewing if we exclude 8

links-loops depicting the reflexivity of the relation (diagonal elements of the matrix) and transitive links, i.e. bundles

If there is an intermediate number Z such that

(for example, a bunch because ). Then in the schema

there will be 12 ligaments (Fig. 15); the missing links are implied "by transitivity". The number 1 is the smallest and the number 30

the largest elements in . If we exclude from the number 30 and

consider the same partial order on the set , then

there is no largest element, but there are 3 maximum elements: 6, 10, 15

Now let's build the same scheme for the Boolean relation

(set of all subsets) of a three-element set

Contains 8 elements:

Check that if you match the elements a, b, c, the numbers 2, 3, 5, respectively, and the operations of union of sets are the multiplication of the corresponding numbers (i.e., for example, a subset corresponds to

product 2 5 = 10), then the relation matrix will be exactly

same as for relation ; schemes of these two relations with the described

abbreviations of loops and transitive connectives coincide up to notation (see Fig. 16). The smallest element is

And the biggest -

binary relations R on the set A And S on the set IN called isomorphic if between A and B it is possible to establish a one-to-one correspondence Г, in which, if (i.e.

elements are related R), then (images

these elements are related S).

Thus, partially ordered sets and are isomorphic.

The considered example admits a generalization.

The Boolean relation is a partial order. If

Those. a bunch of E contains P elements , then each

subset corresponds P-dimensional vector with

components , where is the characteristic function

sets A/ . The set of all such vectors can be considered as a set of points P-dimensional arithmetic space with coordinates 0 or 1, or, in other words, as vertices P-dimensional

unit cube, denoted by , i.e. cube with edges of unit length. For n = 1, 2, 3 indicated points represent respectively the ends of the segment, the vertices of the square and the cube - hence the common name. For /7=4, a graphic representation of this relationship is in Fig.17. Near each vertex of the 4-dimensional cube, the corresponding

subset of a 4-element set and four-dimensional

a vector representing the characteristic function of this subset. The vertices are connected to each other, corresponding to subsets that differ in the presence of exactly one element.

In Fig. 17, a four-dimensional cube is depicted in such a way that on one

level there are pairwise incomparable elements containing the same number of units in the record (from 0 to 4), or, in other words, the same number of elements in the represented subsets.

In Fig.18a,b - other visual representations of a 4-dimensional cube;

in Fig.18a the axis of the first variable OH directed upwards (intentional deviation from the vertical so that the various edges of the cube do not merge):

while the 3-dimensional subcube corresponding to X= 0 is located below, and for X= 1 - higher. On fig. 186 same axle OH directed from inside the cube to the outside, the inner subcube corresponds to X= Oh, and external - X= 1.

IN
The material file shows an image of a 5-dimensional unit cube (p. 134).

Relationship properties:

1) reflexivity;

2) symmetry;

3) transitivity.

4) connectedness.

Attitude R on the set X called reflective if about each element of the set X it can be said that it is in relation R With myself: XRx. If the relation is reflexive, then there is a loop at each vertex of the graph. Conversely, a graph whose every vertex contains a loop is a reflexive relation graph.

Examples of reflexive relations are the “multiple” relation on the set of natural numbers (each number is a multiple of itself), and the similarity relation of triangles (each triangle is similar to itself), and the “equality” relation (each number is equal to itself), etc.

There are relations that do not have the property of reflexivity, for example, the relation of perpendicularity of segments: ab, ba(there is no segment that can be said to be perpendicular to itself) . Therefore, there are no loops on the graph of this relation.

It does not have the property of reflexivity and the ratio is “longer” for segments, “greater by 2” for natural numbers, etc.

Attitude R on the set X called anti-reflexive, if for any element from the set X always false XRx: .

There are relations that are neither reflexive nor antireflexive. An example of such a relationship is the relationship "point X symmetrical to a point at relatively straight l”, defined on the set of points of the plane. Indeed, all points of the line l are symmetrical to themselves, and points that do not lie on a line l, are not symmetrical to themselves.

Attitude R on the set X called symmetrical, if the condition is met: from the fact that the element X is in relation to the element y, it follows that the element y is in relation R with element X:xRyyRx .

The graph of a symmetric relation has the following feature: together with each arrow coming from X To y, the graph contains an arrow going from y To X(Fig. 35).

Examples of symmetrical relationships can be the following: the ratio of "parallelism" of segments, the ratio of "perpendicularity" of segments, the ratio of "equality" of segments, the ratio of similarity of triangles, the ratio of "equality" of fractions, etc.

There are relations that do not have the property of symmetry.

Indeed, if the segment X longer than segment at, then the segment at cannot be longer than the segment X. The graph of this relation has a singularity: the arrow connecting the vertices is directed only in one direction.

Attitude R called antisymmetric, if for any elements X And y out of truth xRy falsity follows yRx: : xRyyRx.

In addition to the “longer” relation, there are other antisymmetric relations on the set of segments. For example, the relation "greater than" for numbers (if X more at, That at can't be more X), the ratio “more by”, etc.

There are relations that have neither the property of symmetry nor the property of antisymmetry.

Relation R on the set X called transitive if from what element X is in relation R with element y, and the element y is in relation R with element z, it follows that the element X is in relation R with element z: xRy And yRzxRz.

Graph of a transitive relation with each pair of arrows going from X To y and from y To z, contains an arrow going from X To z.

The relation "longer" on the set of segments also has the property of transitivity: if the segment A longer than segment b, line segment b longer than segment With, then the segment A longer than segment With. The "equality" relation on the set of segments also has the property of transitivity: (a=b, b=c)(a=c).

There are relations that do not have the property of transitivity. Such a relation is, for example, the perpendicularity relation: if the segment A perpendicular to the segment b, and the segment b perpendicular to the segment With, then the segments A And With not perpendicular!

There is another property of relations called the connected property, and a relation that has it is called connected.

Attitude R on the set X called related, if for any elements X And y from this set, the following condition is satisfied: if X And y are different, then either X is in relation R with element y, or element y is in relation R with element X. Using symbols, this can be written like this: xyxRy or yRx.

For example, the relation "greater than" for natural numbers has the property of being connected: for any different numbers x and y, one can assert either x>y, or y>x.

In a relation graph, any two vertices are connected by an arrow. The converse is also true.

There are relations that do not have the connectedness property. Such a relation, for example, is the relation of divisibility on the set of natural numbers: we can call such numbers x and y whatever the number X is not a divisor y, no number y is not a divisor X(numbers 17 And 11 , 3 And 10 etc.) .

Let's look at a few examples. On the set X=(1, 2, 4, 8, 12) the relation "number X multiple of y". Let us construct a graph of this relation and formulate its properties.

They say about the relation of equality of fractions, it is an equivalence relation.

Attitude R on the set X called equivalence relation, if it simultaneously possesses the properties of reflexivity, symmetry and transitivity.

Examples of equivalence relations are: equality relations geometric shapes, the ratio of parallelism of lines (provided that coinciding lines are considered parallel).

In the relation of "equality of fractions" discussed above, the set X split into three subsets: ; ; }, {; } , (). These subsets do not intersect, and their union coincides with the set X, i.e. we have a partition of the set into classes.

So, if an equivalence relation is given on a set X, then it generates a partition of this set into pairwise disjoint subsets - equivalence classes.

Thus, we have established that the relation of equality on the set
X=( ;; ; ; ; ) corresponds to the partition of this set into equivalence classes, each of which consists of equal fractions.

The principle of dividing a set into classes by means of some equivalence relation is an important principle of mathematics. Why?

First, equivalent - it means equivalent, interchangeable. Therefore, elements of the same equivalence class are interchangeable. So, fractions that are in the same equivalence class (; ; ), are indistinguishable from the point of view of the relation of equality, and the fraction can be replaced by another, for example . And this replacement will not change the result of the calculations.

Secondly, since there are elements in the equivalence class that are indistinguishable from the point of view of some relation, it is believed that the equivalence class is determined by any of its representatives, i.e. arbitrary element of the class. So, any class of equal fractions can be specified by specifying any fraction belonging to this class. An equivalence class with respect to one representative allows, instead of all elements of a set, to study a set of representatives from equivalence classes. For example, the equivalence relation "have the same number of vertices" given on a set of polygons generates a partition of this set into classes of triangles, quadrangles, pentagons, and so on. properties inherent in a certain class are considered on one of its representatives.

Thirdly, the division of a set into classes with the help of an equivalence relation is used to introduce new concepts. For example, the concept of "a bundle of lines" can be defined as that common thing that parallel lines have with each other.

Other important view relations are relations of order. Consider the problem. On the set X={3, 4, 5, 6, 7, 8, 9, 10 ) the relation “have the same remainder when divided by 3 ". This relation generates a partition of the set X into classes: one will include all numbers, when divided by 3 obtained in the remainder 0 (these are numbers 3, 6, 9 ). In the second - numbers, when divided by 3 the remainder is 1 (these are numbers 4, 7, 10 ). The third will include all numbers, when divided by 3 the remainder is 2 (these are numbers 5, 8 ). Indeed, the resulting sets do not intersect and their union coincides with the set X. Therefore, the relation "to have the same remainder when divided by 3 » defined on the set X, is an equivalence relation.

To take another example, many students in a class can be sorted by height or age. Note that this relation has the properties of antisymmetry and transitivity. Or everyone knows the order of the letters in the alphabet. It is provided by the “should” attitude.

Attitude R on the set X called strict order relation, if it simultaneously possesses the properties of antisymmetry and transitivity. For example, the relation X< y».

If the relation has the properties of reflexivity, antisymmetry and transitivity, then such it will be non-strict order relation. For example, the relation Xy».

Examples of the order relation are: the relation “less than” on the set of natural numbers, the relation “shorter” on the set of segments. If an order relation also has the property of being connected, then it is said to be linear order relation. For example, the relation "less than" on the set of natural numbers.

A bunch of X called orderly, if it has an order relation.

For example, many X={2, 8, 12, 32 ) can be ordered using the “less than” relation (Fig. 41), or this can be done using the “multiple” relation (Fig. 42). But, being an order relation, the relations “less than” and “multiply” order the set of natural numbers in different ways. The "less than" relation allows you to compare any two numbers from the set X, and the relation "multiply" does not have such a property. Yes, a couple of numbers. 8 And 12 is not bound by the relation "multiply": it cannot be said that 8 multiple 12 or 12 multiple 8.

It should not be thought that all relations are divided into equivalence relations and order relations. There are a huge number of relations that are neither equivalence relations nor order relations.

Lecture Plan #14 Classification of binary relations

1. Classification of antisymmetric relations
2. Classification of reflexive relations
2.1. Quasi-order relations
2.2. Relations of non-strict partial order
2.3. Non-strict ordering relations
2.4. Poor quality order
2.5. Non-strict weak order
2.6. Non-strict order
3. Duality of relations of strict and non-strict order
4. Overview of properties various kinds relations

Classification of antisymmetric relations

Structure of graphs of acyclic relations

The structure of graphs of relations of qualitative order

Structure of relation graphs of weak order

Strict order relationships

A strict order (strict preference, strong order, strict linear order) is an antireflexive, transitive, weakly connected binary relation (12).

Strict order is a special case of weak order (strict partial preference) with an additional weakly connected condition.

Example: The relation "strictly less than" on the set of integers.

Classification of reflexive relations

Quasi-order relations

These binary relations make it possible to compare elements of a certain set, but not by similarity, but by arranging the elements of groups in a certain order, i.e. by partial ordering.

A quasi-order (non-strict partial preference) is a reflexive and transitive binary relation (3).

Example: "to be a brother" (Ivan-Peter, Andrey-Anna)

Properties of quasi-orders

1. The intersection of quasi-orders remains a quasi-order.
2. The symmetric part of the quasi-order has the properties of reflexivity, symmetry, and transitivity, and therefore is an equivalence relation. R c = R / R inv
3. With the help of this intersection, it is possible to select groups of variants that are equivalent to each other, then a non-strict partial order relation generated by the original relation can be established between the selected groups.
4. The asymmetric part of the quasi-order is a transitive and anti-reflexive relation = qualitative order.

Relations of non-strict partial order

A nonstrict partial order relation (4) is a relation that has the properties of reflexivity, antisymmetry, and transitivity.

A non-strict partial order is an antisymmetric quasi-order

Example: "be part" relation defined for sets (and their subsets)

Properties of non-strict partial orders

1. The intersection of nonstrict partial orders remains a nonstrict partial order.
2. The symmetric part of a nonstrict partial order is a diagonal.
3. The asymmetric part of a nonstrict partial order is a (strict) qualitative order.
4. In the theory of intelligent systems, an important role is played by partially ordered sets - domains together with non-strict partial order relations defined on them.
5. Partially ordered sets with the additional property that every pair of elements has upper and lower bounds are called lattices. Boolean algebras are a particular case of lattices.

Non-strict ordering relationships

A nonstrict ordering is a reflexive relation that has the weakly connected property (5).

A loose ordering can also be defined as a fully connected relation.

The non-strict ordering relation can be thought of as the result of combining some tolerance and dominance relations.

Properties of relations of non-strict partial ordering

1. The intersection and union of fully connected relations remains a fully connected relation.
2. The symmetric part of the non-strict partial ordering is the tolerance.
3. The asymmetric part of a nonstrict partial ordering is a dominance.
4. For fully connected relations, a necessary condition for transitivity is that the relation is negatively transitive.
5. For fully connected relations, the property of transitivity is a sufficient condition for the relation to be negatively transitive.

Relations of nonstrict qualitative order

A binary relation R is called a nonstrict qualitative order if it is negative and fully connected (6).

A nonstrict qualitative order is a negative nonstrict ordering.

The relation of nonstrict qualitative order can be represented as the result of combining some relations of tolerance and qualitative order.

Properties of relations of nonstrict qualitative order

1. The symmetrical part of the non-strict qualitative order is tolerance. NT?
2. The asymmetric part of a non-strict qualitative order is transitive, and therefore is a qualitative order relation.
3. Thus, the non-strict qualitative order relation can be represented as the result of the union of the tolerance and qualitative order relations generated by the original relation.
4. The dual relation has the properties of asymmetry and transitivity, therefore it is a relation of a qualitative order.

Relations of non-strict weak order

A nonstrict weak order is a fully connected transitive and negative transitive relation (7).

A nonstrict weak order is a fully connected transitive relation.

A non-strict weak order is a transitive non-strict order.

Properties of relations of non-strict weak order

1. The symmetric part of a nonstrict weak order is an equivalence.
2. The asymmetric part Rac of a nonstrict weak order is transitive, and therefore is a relation of qualitative order.
3. Thus, a non-strict weak order relation can be represented as the result of the union of the equivalence and weak order relations generated by the original relation.
4. A non-strict weak order can be represented as a set of partially ordered layers, each of which is an equivalence class.

Relations of non-strict (linear) order

A non-strict order (non-strict linear order) is an antisymmetric, transitive, fully connected binary relation (8).

A non-strict order is an antisymmetric non-strict weak order.

A non-strict order is an anti-symmetric non-strict order.

Properties of relations of non-strict linear order

1. The symmetric part of a non-strict order is a diagonal.
2. The asymmetric part R ac of non-strict order is transitive and weakly connected, and therefore is a relation of strict order.
3. The dual relation has the properties of asymmetry, negativity and weakly connectedness; therefore, it is a relation of strict order. In addition, it coincides with R ac.
4. Thus, the non-strict order relation can be represented as the result of the union of the diagonal and the strict order generated by the original relation.

Duality of relations of strict and nonstrict order

An overview of the properties of different types of relationships

The word "order" is often used in a variety of issues. The officer gives the command: “Calculate in order of numbers”, arithmetic operations are performed in a certain order, athletes become in height, there is an order for performing operations in the manufacture of a part, word order in a sentence.

What is common in all cases when it comes to order? The fact that the word “order” has such a meaning: it means which element of this or that set follows which (or which element precedes which).

Attitude " X follows at» transitively: if « X follows at" And " at follows z", That " x follows z". In addition, this ratio must be antisymmetric: for two different X And at, If X follows at, That at does not follow X.

Definition. Attitude R on the set X called strict order relation, if it is transitive and antisymmetric.

Let us find out the features of the graph and the graph of strict order relations.

Consider an example. On the set X= (5, 7, 10, 15, 12) the relation R: « X < at". We define this relation by enumeration of pairs
R = {(5, 7), (5, 10), (5, 15), (5, 12), (7, 10), (7, 15), (7, 12), (10, 15), (10, 12), (12, 15)}.

Let's build its graph. We see that the graph of this relation has no loops. There are no double arrows on the graph. If from X the arrow goes to at, and from at- V z, then from X the arrow goes to z(Fig. 8).

The constructed graph allows you to arrange the elements of the set X in this order:

{5, 7, 10, 12, 15}.

In Fig. 6 (§ 6 of this chapter) columns VII, VIII are graphs of relations of a strict order.

Non-strict order relation

The relation "less than" in the set of real numbers is opposite to the relation "not less". It is no longer a strict order. The point is, at X = at, relations X ³ at And at ³ X, i.e. the relation "no less" is reflexive.

Definition. Attitude R on the set X called non-strict order relation, if it is reflexive, antisymmetric and transitive.

Such relations are unions of a strict order relation with an identity relation.

Consider the relation "no more" (£) for the set

X= (5, 7, 10, 15, 12). Let's build its graph (Fig. 9).

A non-strict order relation graph, unlike a strict order relation graph, has loops at each vertex.

On fig. 6 (§ 6 of this chapter) graphs V, VI are graphs of relations of non-strict order.

Ordered sets

A set may turn out to be ordered (they also say completely ordered) by some order relation, while another may be unordered or partially ordered by such a relation.

Definition. A bunch of X called orderly some order relation R if for any two elements x, y from X:

(X, at) Î R or ( y, x) Î R.

If R is a strict order relation, then the set X ordered by this relation under the condition: if X, at any two unequal elements of a set X, That ( X, at) Î R or ( y, x) Î R, or any two elements x, y sets X are equal.

It is known from the school mathematics course that number sets N , Z , Q , R ordered by the ratio "less than" (<).

The set of subsets of a certain set is not ordered by the introduction of an inclusion relation (U), or a strict inclusion relation (T) in the above sense, because there are subsets none of which is included in the other. In this case, the given set is said to be partially ordered by the relation Í (or Ì).

Consider the set X= (1, 2, 3, 4, 5, 6) and it has two relations "less than" and "divisible by". It is easy to check that both of these relations are order relations. The less than relation graph can be represented as a ray.

The graph of the relation "is divided by" can be represented only on a plane.

In addition, there are vertices on the graph of the second relation that are not connected by an arrow. For example, there is no arrow connecting the numbers 4 and 5 (Fig. 10).

The first relation X < at' is called linear. In general, if the order relation R(strict and non-strict) on the set X has the property: for any X, atÎ X or xRy, or yRx, then it is called a linear order relation, and the set X is a linearly ordered set.

If the set X of course, and consists of n elements, then the linear ordering X reduces to enumeration of its elements by numbers 1,2,3, ..., n.

Linearly ordered sets have a number of properties:

1°. Let a, b, c– set elements X, ordered by relation R. If it is known that aRv And vRc, then we say that the element V lies between the elements A And With.

2°. A bunch of X, linearly ordered by the relation R, is called discrete if between any two of its elements lies only a finite set of elements of this set.

3°. A linearly ordered set is called dense if for any two distinct elements of this set there is an element of the set lying between them.

Equivalence relation. Connection of the equivalence relation with the division of a set into classes

Definition. Attitude R on the set X is called an equivalence relation if it is reflexive, symmetric, and transitive.

Example. Consider the relationship " X classmate at» on a set of students of the pedagogical faculty. It has properties:

1) reflexivity, since each student is a classmate to himself;

2) symmetry, because if student X at, then the student at is a classmate of a student X;

3) transitivity, because if student X- classmate at, and the student at- classmate z, then the student X be a classmate of a student z.

Thus, this relation has the properties of reflexivity, symmetry and transitivity, and therefore is an equivalence relation. At the same time, the set of students of the pedagogical faculty can be divided into subsets consisting of students enrolled in the same course. We get 5 subsets.

The equivalence relation is also, for example, the relation of parallel lines, the relation of equality of figures. Each such relation is connected with the division of the set into classes.

Theorem. If on the set X given an equivalence relation, then it splits this set into pairwise disjoint subsets (equivalence classes).

The converse statement is also true: if any relation defined on the set X, generates a partition of this set into classes, then it is an equivalence relation.

Example. On the set X= (1; 2; 3; 4; 5; 6; 7; 8) the relation "have the same remainder when divided by 3" is given. Is it an equivalence relation?

Let's build a graph of this relation: (independently)

This relation has the properties of reflexivity, symmetry and transitivity, therefore, it is an equivalence relation and splits the set X into equivalence classes. Each equivalence class will have numbers that, when divided by 3, give the same remainder: X 1 = {3; 6}, X 2 = {1; 4; 7}, X 3 = {2; 5; 8}.

It is believed that the equivalence class is determined by any of its representatives, i.e. arbitrary element of this class. So, the class of equal fractions can be specified by specifying any fraction belonging to this class.

In the initial course of mathematics, equivalence relations also occur, for example, "expressions X And at have the same numerical values", "figure X equal to figure at».

Definition. Attitude R on the set X is called an order relation if it is transitive and asymmetric or antisymmetric.

Definition. Attitude R on the set X is called a strict order relation if it is transitive and asymmetric.

Examples relations of a strict order: "greater" on the set of natural numbers, "higher" on the set of people, etc.

Definition. Attitude R on the set X is called a relation of nonstrict order if it is transitive and antisymmetric.

Examples relations of a non-strict order: “no more” on the set of real numbers, “to be a divisor” on the set of natural numbers, etc.

Definition. A bunch of X is called ordered if an order relation is given on it.

Example. On the set X= (1; 2; 3; 4; 5) two relations are given: " X £ at" And " X- divider at».

Both of these relations have the properties of reflexivity, antisymmetry, and transitivity (build the graphs and check the properties yourself), i.e. are a relation of non-strict order. But the first relation has the property of being connected, while the second does not.

Definition. Order relation R on the set X is called a linear ordering relation if it has the property of being connected.

Many relationships of order are studied in elementary school. Already in the first class, there are relations “less”, “greater” on the set of natural numbers, “shorter”, “longer” on the set of segments, etc.

Control questions

1. Define a binary relation on a set X.

2. How to write a statement that the elements X And at are in relation R?

3. List ways of setting relationships.

4. Formulate the properties that relationships can have. How are these properties reflected in the graph?

5. What properties must a relation have in order for it to be an equivalence relation?

6. How is the equivalence relation related to the division of a set into classes?

7. What properties must a relation have in order for it to be an order relation?