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One of the basic concepts for mathematics is sets. A set is defined as the grouping of various variables into a single entity, mostly within a curly bracket. These seats are coordinated with another set on the basis of several conditions, and this is termed as the relations. Relations determine the link and coordination between different sets on the basis of their properties, equations and evaluations. There are a number of relations that are determined between different sets, which includes, universal relation, empty relation, inverse relation, identity relation, etc.
Relation
A relation is a link between the elements of two or more sets. Sets must not be empty. If two sets are investigated, the relationship between them will be confirmed if the elements of two or more sets are related. The subset representation of any Cartesian product is represented by the set or a relation R. The Roster or Set-builder methods can both be used to express a relation.
Sets and Relations
Sets and relationships are inextricably linked. The relationship established between two distinctive sets is defined by the relation.
Whether two sets are accessible, we may use relations to see if there is any relationship between them.
An empty relation, for example, means that no two elements in the two sets are the same
Relations in Mathematics.
The connection is the relationship established between two or more sets of values in mathematics.
Assume that x and y are two ordered pair sets. If set x has a relationship with set y, the values of set x are referred to as domain, while the values of set y are referred to as range.
Types of Relations
- Empty Relation (Void Relation)
The relation is termed as an empty relation only if there are no coordinates or relation between the elements of the sets. If set A = {3, 6, 9} then an empty relation can be represented by set B = {x, y} where |x + y| = 2.
- Universal Relation
The universal relation is one in which every element of a certain set is connected to one another. Consider the set A = {p, q, r} and B = {a, b} where |a-b| 0. Here the relationship between Set A and Set B represents a universal relation.
- Identity Relation
When the elements of the set are related to each other, i.e., related to themselves, the relation is defined as the identity relation. The identity relation in a set A = {p, q, r} therefore the relation will be represented as I ={p, p}, {q, q}, {r, r}. In terms of the identity relationship.
- Inverse Relation
When a set consists of an inverse pair of the elements of other sets, the whole relation will be termed as an inverse relation. For example, if set A = {(p, q), (b, c)}, then inverse relation will be I-1= {(q, p), (b, c)}.
- Reflexive Relation
Every element in a reflexive relationship is considered to map itself. Assuming, the set A = {a, b}, then R = {(a, a), (b, b), (a, b), (b, a)} is an example of a reflexive connection.
- Symmetric Relation
If any set is said to be in the symmetric relation, then the evaluation is in the symmetric form, i.e., if p=q, then q=p. To put it another way, a relation R is only symmetric if (q, p) R is true when (p,q) R.
R = (1, 2), (2, 1) for a set A = 1, 2 is an example of symmetric relation.
- Transitive Relation
The transitive relation can be represented as (a, b) ∈ R, (b, c) ∈ R, then (c, a) ∈ R.
- Equivalence Relation
The relation that is termed to have the property of the symmetric, reflexive, as well as a transitive relation, is said to be Equivalence Relation.
Representation of Types of Relationships
Relation Type | Condition |
|---|---|
Void Relation | R = φ ⊂ A × A |
Universal Relation | R = A × A |
Identity Relation | I = {(p, q), p ∈ A} |
Inverse Relation | R-1 = {(p, q): (p, q) ∈ R} |
Reflexive Relation | (p, p) ∈ R |
Symmetric Relation | pRq ⇒ qRp, ∀ p, q ∈ A |
Transitive Relation | pRq and qRr ⇒ pRr ∀ p, q, r ∈ A |
Representation of Relations in Mathematics
In mathematics, the relation can be expressed in 3 different categories:
- Roster Form
In the Rooster form, the set consists of both the values of the first component and the second component. However, the evaluation of greater than, less than or multiplication is applied between the components. The symbol used is <, >, =, etc.
For example,
1. If A = {p, q, r} B = {7, 8, 9}
then R = {(p, 7), (q, 8), (r, 9)}
Hence, R A*B
- Given A = {2, 3, 6, 10} B = {4, 2, 7, 1} therefore, the relationship is being the set A and set B, is termed as, “is less than” and they are represented in the Roster form as R = {(2, 4) (2, 7) (3, 4), (3, 7), (6, 7)}
Here, the value of the first component is less than that of the second component.
If A = {-2, 2, 1} and B = {1, 9, 4, 10}
If the Roster form is represented as (a² = b)
Therefore, R = {(-2, 4), (2, 4), (1, 1)
- Set Builder form
In the site builder form, the set A and B are represented by R = {(p, q): p ∈ A, q ∈ B, p…q}. Moreover, the rules associated with the values of p and q might determine the further representation.
For Example:
Let R = {(4, 6), (6, 8), (8, 10)}
In the given representation of the set builder form, the value of a is two less than that of the value of a b.
- Arrow Diagram
The Arrow diagram is the representation of sets A and B in the form of a circle. The circle is named A and B, where each circle represents the set an A and B, respectively. The values of each set are placed in their respective circles, and an arrow is used to join the values that correspond to each other or share equal values.
For Example,
If A = {2, 5, 4} B = {2, 6, 4, 9, 25, 15, 16 }, If the arrow diagram is used for representing the positive square root of the set A in correspondence to set B, then the representation will be,
R = {(2, 4); (5, 25); (4, 16)}
Here, two circles are drawn with values of set A in the circle A and the value of B in the circle B. An Arrow is used to join the values of the first component with the second component, which is the perfect square root of the former.
Conclusion:
A relation is defined as the property of two distinctive sets and how they both are quite coordinated, related, or linked to each other. On the basis of various distinctions, the relations are categorised into eight basic groups, which include universal relation, inverse relation, reflexive relation, identity relation, etc. Moreover, these relations between the sets are represented physically. The category of representation of the relations in mathematics is booster form, set builder form, and arrow diagram. Every relation follows a particular condition for determining the relationship between two sets.
