Reflexivity is a desire for maximum awareness of one’s own actions, in other words, a determination to find the meaning of ongoing events and the relationships between them. It is considered a system-forming and multifunctional personal quality that contributes to the effective assimilation, deepening and processing of social experience, switching from the external level to the internal plane.
Reflexivity is also the ability of individuals and society to think critically, see negative aspects, pathological phenomena, identify potential dangers, and take preventative measures that can prevent or slow down certain trends.
Properties of relationships
Reflexivity . A relation R defined on a set M is called reflexive if for any relation it holds. Formally, reflexivity can be defined as follows: A reflexive relation always holds between the object and itself. The most striking examples of reflexive relationships are self-care and equality.
Anti-reflexivity. A relation R defined on a set M is anti-reflexive if In anti-reflexive relations it follows from the condition that Examples of anti-reflexive relations: to be older, to be smaller, etc.
Symmetry. A relation R defined on a set M is called symmetric if, when the relation is satisfied, the relation is also satisfied. Formally, the relation is symmetric if, for example, the relations “to stand next to each other on a shelf” on a set of books or “to be relatives” on a set of people are symmetrical.
Asymmetry . A relation R, defined on a set M, is called asymmetric if in asymmetric relations of two relations and no more than one (one or not one) can be satisfied. An example of an asymmetrical relationship: “to be a father” (if he is a father, then he cannot be a father).
Antisymmetry . A relation R defined on a set M has the property of antisymmetry if This means that if the relations are simultaneously satisfied, all relations of a non-strict order are antisymmetric: “to be no more”, “to be no higher”, etc.
Transitivity. A relation R defined on a set M is transitive if for any of the satisfiability conditions
relations follows Formally, this can be written as follows: A relation that does not have such a property is said to be intransitive. For example, the relation “standing next to each other on a shelf” is intransitive. Indeed, let the volumes of some encyclopedia be in ascending order of volume numbers. Then, if the first volume stands next to the second, and the second next to the third, then, obviously, the first does not stand next to the third.
All general properties of relations can be divided into three groups: reflexivity (each relation is reflexive or anti-reflexive), symmetry (a relation is always either symmetrical, or asymmetrical, or antisymmetrical), transitivity (every relation is transitive or non-transitive). Relations that have a certain set of properties are given special names.
There are relationships that do not have the property of reflexivity.
Examples of relationships that do not have the property of reflexivity:
- the relation of perpendicularity on a set of segments (there is not a single segment about which it can be said that it is perpendicular to itself);
— the “longer” relation for segments.
Definition.
A relation
R on a set A is called anti-reflexive if it can be said about each element of the set A that it is not in a relation R with itself, that is, aRa
.
Definition.
A relation
R
on a set
A is called symmetric if the condition is met: from the fact that element a
is in a relation
R
with an element
b
, it follows that the element
b
is in a relation
R
with an element
a
, that is, for any of
aRb
it follows
bRa
.
Comment. The symmetric relation graph has a special feature: together with each arrow coming from a
to b, the graph also contains an arrow going from
b
to a.
The opposite statement is also true. A graph containing each arrow going from a
to
b
and an arrow going from
b
to
a
is a symmetric relation graph.
Examples of symmetrical relationships:
- parallelism relation on a set of lines (if line x
is parallel to the line
y
, then the line
y
is parallel to the line
x
);
— similarity ratio of triangles (if triangle F
is similar to triangle
P
, then triangle
P
is similar to triangle
F
).
- the relationship of perpendicularity on a set of segments (if one segment is perpendicular to another segment, then this “other” is perpendicular to the first);
— the “longer” relation for segments (if one segment is equal to another segment, then this “other” is equal to the first).
EXAMPLE. Let us consider the relation “longer” on a set of segments, which does not have the property of symmetry. Indeed, if the segment x
is longer than the segment
y
, then the segment
y
cannot be longer than the segment
x
. About the relationship “longer” they say that it has the property of antisymmetry or is simply antisymmetric.
Definition.
A relation
R
on a set
A is called antisymmetric if for different elements a
and
b
from the set
A the following condition is satisfied: if aRb
and
bRa
imply
a
=
b
.
Comment. An antisymmetric relation graph has a special feature: if the vertices of the graph are connected by an arrow, then there is only one arrow. The converse is also true: a graph whose vertices are connected by only one arrow is an antisymmetric relation graph.
Examples of antisymmetric relationships:
— the relation “longer” on a set of segments;
- “greater than” relation for numbers (if x
greater
than y
, then
y
cannot be greater than
x
);
- the relation “more than 2” for numbers (if x
is greater than y by 2, then y cannot be greater than
x
).
EXAMPLE. Let us consider the relation “to be a sister” on the set of children of one family, which has neither the property of symmetry nor the property of antisymmetry. Let there be three children in the family: Katya, Masha, and Tolya. Then the graph of the relationship “being a sister” will be like this:
K··M
·
T
Figure 3.3 – Relationship graph “Being a sister”
It shows that this relation has neither the property of symmetry nor the property of antisymmetry.
Definition.
A relation
R
on a set
A is called transitive if the condition is met: from the fact that element a
is in a relation
R
with element
b, and
element
b
is in a relation
R
with element
c
, then it follows that element
a
is in relation
R
with element c, that is, for any of
a
R
b
and
b
R
c,
a
R
follows .
Comment. Transitive relation graph with each pair of arrows coming from a
to
b
and
b
to
c
, contains an arrow going from
a
to
c
. The opposite statement is also true.
EXAMPLE. The relation ≤ on the set R of real numbers, as well as the inclusion relation of subsets of a certain set, are reflexive and transitive, but not symmetric. The relation < on the set of real numbers is transitive, but not reflexive and not symmetric. Relation " x"
is the mother of y” is not reflexive, not symmetrical, not transitive.
Let us now consider the properties of binary relations in the language of matrices.
Let R
– a binary relation on the set.
Ratio
: _
· reflexively, if only ones are located on the main diagonal of the relation matrix;
· symmetrically, if the matrix is symmetrical with respect to the main diagonal;
· antisymmetric if in the matrix all elements outside the main diagonal are zero;
· transitive if the relation is satisfied.
EXAMPLE. Let's check what properties the relation , A
={1,2,3}, defined by the relation graph.
Let's create a relation matrix R:
Since the matrix has zero elements on the main diagonal, the relation R is not reflexive
.
Matrix asymmetry means that the relation R is not symmetrical.
To check the antisymmetry, let's calculate the matrix.
Since in the resulting matrix all elements outside the main diagonal are zero, the ratio R is antisymmetric
.
Since (check!), then, that is, R
is
a transitive
relation.
Equivalence relation
A relation R that has the properties of reflexivity, symmetry and transitivity is called an equivalence relation . For equivalent relations, the notation is usually written (read: "equivalent"). Equivalent relations are: “to be congruent” on a set of flat triangles, “to be the same size” on a set of shoe samples, “to be related” on a set of people, etc.
The introduction of an equivalence relation R on a set M determines the partition of all elements of this set into equivalence classes. The set of all equivalence classes forms the factor set of the set M and is denoted M/R. Moreover, each element of this class is an authorized representative of this class. A set of one and only one representative of each class is called a system of representatives of the corresponding equivalence relation. An example of the introduction of an equivalence relationship and the formation of a system of representatives is the formation of a representative body of power based on elections.
What it is
In order to understand the concept of reflexivity, it is necessary to clarify the meaning of the word reflection, which implies the process of individuals turning attention inward and to their own consciousness, then reflexivity means the ability of subjects to comprehend their personality, to understand the origins of their own actions.
Thus, reflection is a qualitative characteristic indicating that consciousness is capable of directing attention inward. Reflexivity is a quantitative characteristic demonstrating the severity of such ability, criticality of analysis, and depth.
Reflection literally means going back. The concept of reflecting is used to denote the process of thinking about one’s own mental state, turning attention inward. In different sciences and philosophical views, the term in question has different meanings. Thus, Locke meant by reflection the source of specific knowledge when the action of consciousness is contemplated. Leibniz considered reflection to be the attention of individuals to what is happening within them. According to Jung, ideas are reflections on ideas received from the outside.
Summarizing all views, what is common in the interpretation of reflection is its focus on the inner world of individuals.
Reflexivity acts as a personal trait that characterizes self-directed cognition. The analyzed phenomenon is considered one of the most important characteristics of consciousness.
Psychologists emphasize the importance of separating the two similar phenomena above. Reflexivity in psychology is presented as a personal property, and reflection is a process of knowing one’s own personality.
In this case, the first refers to the quality of personality that determines reflection as a process. Reflexivity has a criterion of expression and is determined by the direction of the processes of cognition inside one’s own personality.
These two above phenomena are realized with the help of reflexive abilities.
Thus, reflexivity is the ability to analyze one’s own personality, discover the motives of one’s actions, including:
– past actions and events;
- emotional condition;
– successful or unsuccessful performance results;
– changing personality traits, character traits.
Each individual has a different level of reflection. It is determined by intellectual development, level of knowledge, and upbringing. Some subjects continuously analyze their own actions, reflect on the motivation of their actions, while others do not think about their own behavior at all. A huge role here is given to the individual’s desire to understand his own misconceptions, to realize mistakes, the level of self-criticism and the need to compare his own person with the environment.
The distinguishing feature of a mature personality is the ability to be responsible for committed actions and take responsibility for the existence that a person has. If an individual invariably blames the environment and circumstances for current events and current situations, then he is a weak person.
Reflexivity of consciousness is the ability to comprehend one’s own internal “Universe” and build a picture of personal states.
General consciousness performs the function of self-awareness (the ability to isolate one’s own person as a separate creation) with interaction and orientation to the external environment. It is inherent in all beings. While reflexive consciousness is responsible for self-knowledge (the ability to analyze the state of one’s own spiritual world), it is developed during development as a specific feature inherent exclusively in human consciousness.
The immediate possibility of reflection in individuals arises as a result of knowledge of the environment and, in particular, society. Mastery of historically developed operations of activity and methods of interaction with the environment gives rise to the need for reflection, and, consequently, leads to the emergence of a perfect form of reflection of reality. Only at the level of the perfect form of reflection does reflexivity of consciousness arise.
Attitude of tolerance
A relation defined on a set M is called a tolerance relation if it is reflexive, symmetric and intransitive. Notation: If we require transitivity of all pairs of elements from M, we obtain an equivalent relation. Therefore, tolerance can be seen as an extension of equivalence. Equivalence - in the sense of equality, tolerance - in the sense of similarity, similarity. In essence, tolerance means the following: an object is in a given relationship with itself (reflexivity), the similarity of two objects does not depend on the order of comparison (symmetry), but if the first object is similar to the second, and the second is similar to the third, then it is not necessary that the first was similar to the third. Tolerance allows the third, it is not necessary that the first was similar to the third. Tolerance allows you to formalize the intuitive idea of the similarity of objects, their similarity in something. For example, the relation “to be at a distance no more than r”, defined on a set of points on the plane. Rice. Figure 1.15 illustrates this example. Point A is distant from B and C by no more than r, just as point B is from D and C, while A is from D at a distance significantly greater than r.
Order relation
A relation R that has the properties of reflexivity, antisymmetry and transitivity is called an order relation . If an order relation is introduced on a given set, then this set is called ordered. In this case, instead of writing A set is perfectly ordered if for any two elements and from the set M
either , or holds. Otherwise, the set is said to be partially ordered . For example, the relation “to be taller” on the set of trees is perfectly ordered, and the relation “to be a divisor” on the set of integers is partially ordered.
Let each element from the set M, according to some rule f, be assigned a real number, the weight of the element . Entering a weight for each element allows you to order them as their weights increase (decrease) and then compare the elements according to the assigned weight. Examples of ordering through the introduction of weights are: assigning each product its price, each machine its reliability, each body its weight, volume, etc. Weighing decision options through the formation of a complex quality indicator is one of the most common ways to solve the problem of choice based on a variety of different quality attributes.
If a relation has the properties of anti-reflexivity, asymmetry and transitivity, then it is called a relation of strict order (denoted by ). An example of a strict order relation is the order of letters in a fixed alphabet. The ordering of letters in the alphabet allows, in turn, to order words in dictionaries (lexicographic ordering of words).
