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A set is a collection of objects which are termed as elements. The objects of the sets can be anything from people, districts, countries or any other possibilities. A set can be expressed in set builder form or roster form. In roster form, the contents of a set can be described by listing the elements of the set, separated by commas, inside a set of curly brackets. The roster form is also termed as the enumeration form.
Examples of set in roster form:
Write first five natural numbers in roster form: A = {1, 2, 3, 4, 5}
Write vowels in English alphabets in roster form: B = {a, e, i, o, u}
Relation of sets
A relation defines the connection between two sets which is a collection of ordered pairs containing one object from each set. If the object ‘a’ is from the first set and the object ‘b’ is from the second set, then the objects are said to be related if the ordered pair (a, b) is in the relation. The relation of sets can be expressed in set builder form or roster form. The set of x-values is known as the domain, and the set of all y-values is known as the range.
Types of relations
The types of relations are as follows:
Empty Relation
In an empty relation, there is no relation between any elements of a set. [R = φ ⊂ A × A]
Universal Relation
In universal relation, every element of a set is related to each other. [R = A × A].
Identity Relation
In an identity relation, every element of a set is related to itself and no other element. [I = {(a, a), a ∈ A}]
Inverse Relation
In inverse relation, a set is an inverse pair of another set. [R-1 = {(b, a): (a, b) ∈ R}]
Reflexive Relation
In a reflexive relation, every element maps to itself. [(a, a) ∈ R]
Symmetric Relation
In symmetric relation, if a=b is true, then b=a is also true. [aRb ⇒ bRa, ∀ a, b ∈ A]
Transitive Relation
In transitive relation, if (a, b) ∈ R, (b, x) ∈ R, then (a, x) ∈ R. [aRb and bRc ⇒ aRc ∀ a, b, c ∈ A].
Equivalence Relation
If a relation is reflexive, symmetric and transitive at the same time, it is termed as an equivalence relation.
Properties of relations
The properties of relations are as follows:
Reflexivity,
transitivity,
symmetry, and
connectedness
Problems based on relations in roster form
Consider an example of two sets A = {36, 49} and B = {6, 7}. Express the relation between the sets?
The relation is that the elements of A are the square of the elements of B.
The relation in set-builder form is expressed as, R = {(x, y): x is the square of y, x ∈ A and y ∈ B}.
The relation in roster form is expressed as, R = {(36, 6), (49, 7)}.
Consider an example of two sets A = {1, 3, 5} and B = {4, 6, 8}. Express the relation between the sets?
The relation is that the elements of A and B are as follows:
The relation in set-builder form is expressed as, R = {(x, y): y = x + 3}.
The relation in roster form is expressed as, R = {(1, 4), (3, 6), (5, 8)}.
Determine the domain for the relation R = {(1, 4), (3, 6), (5, 8)}.
The set of x-values of a relation is known as the domain. Hence, for the given relation,
Domain = {1, 3, 5}
Determine the range for the relation R = {(1, 4), (3, 6), (5, 8)}.
The set of y-values of a relation is known as the range. Hence, for the given relation,
Range = {4, 6, 8}
Roster form from Venn diagram
The sets and relations expressed in the Venn diagram can also be depicted in the roster form.
The sets from the given Venn diagram can be expressed in roster form as
Set C = {1, 2, 3, 4}
Set D = {5, 6, 7, 8}
The relation of sets can also be expressed in roster form from the Venn diagram.
The relation of the sets C and D is that the elements of set D are the squares of the elements of set A. It can be expressed in roster form as
R = {(1, 1), (5, 25), (9, 81)}
The domain and range of the relation is
Domain = {1, 5, 9} and Range = {1, 25, 81}
Conclusion
The relation between sets is expressed in enumeration form or roster form using curly brackets. Roster form is an easy representation of a set, but a large number of data cannot be represented in roster form.
