Question 1: Show that the relation defined by
is an equivalence relation.
Answer:
To prove that relation is equivalence, we need to prove that it is reflexive, symmetric and transitive.
Reflexivity:
Therefore R is reflexive
Symmetric:
Therefore R is symmetric
Transitive:
Therefore R is transitive
Since, is reflexive, symmetric and transitive,
is an equivalence relation.
Question 2: Show that the relation on the set
of integers, given by
is an equivalence relation.
Answer:
To prove equivalence relation, the given relation should be reflexive, symmetric and transitive.
Reflexivity:
Let be an arbitrary element of the set
.
Therefore, R is reflexive on Z.
Symmetry:
Clearly, R is symmetric on Z
Transitivity:
On adding the above two equations, we get
Thus, R is transitive on Z.
Therefore is reflexive, symmetric and transitive. Hence,
is an equivalence relation on
.
Question 3: Prove that the relation on
defined by
is divisible by 5 is an equivalence relation on
Answer:
To prove equivalence relation, the relation should be reflexive, symmetric and transitive.
Reflexivity:
Let a be an arbitrary element of . Then,
Therefore, R is reflexive on Z.
Symmetry:
Thus, R is symmetric on Z.
Transitivity:
On adding two equations above, we get
Therefore, R is transitive on Z.
Therefore R is reflexive, symmetric and transitive. Hence, R is an equivalence relation on Z.
Question 4: Let be a fixed positive integer. Define a relation
on
as follows:
is divisible by
. Show that
is an equivalence relation on
Answer:
To prove equivalence relation, the relation should be reflexive, symmetric and transitive.
Reflexivity:
Thus, R is reflexive on Z.
Symmetry:
Clearly, R is symmetric on Z.
Transitivity:
Thus, R is transitive on Z.
Therefore R is reflexive, symmetric and transitive. Hence, R is an equivalence relation on Z.
Question 5: Let be the set of integers. Show that the relation
is an equivalence relation on
Answer:
To prove equivalence relation the relation should be reflexive, symmetric and transitive.
Reflexivity:
Thus, R is reflexive on Z.
Symmetry:
Therefore, R is symmetric on Z.
Transitivity:
Adding above two equations, we get
Therefore, R is transitive on Z.
Therefore R is reflexive, symmetric and transitive. Hence, R is an equivalence relation on Z
Question 6: is said to be related to
if
and
are integers and
is divisible by 13. Does this define an equivalence relation?
Answer:
To prove equivalence relation, the given relation should be reflexive, symmetric and transitive.
Reflexivity:
Symmetry:
Therefore, R is symmetric on Z.
Transitivity:
Adding above two equations, we get
So, R is transitive on Z.
Therefore, R is reflexive, symmetric and transitive. Hence, R is an equivalence relation on Z.
Question 7: Let be a relation on the set
of ordered pairs of non-zero integers defined by
Show that
is an equivalent relation.
Answer:
To prove that relation is equivalence, we need to prove that it is reflexive, symmetric and transitive.
Reflexivity :
Therefore R is reflexive.
Symmetric :
Therefore R is symmetric.
Transitive:
Therefore R is transitive.
Since R is reflexive, symmetric & transitive Therefore R is an equivalence relation.
Question 8: Show that the relation on the set
given by
is an equivalence relation. Find the set of all elements related to 1.
Answer:
We have,
Reflexivity:
Therefore R is reflexive.
Symmetric :
Therefore R is symmetric.
Transitive:
Therefore R is transitive.
Since, R is being reflexive, symmetric and transitive, so R is an equivalence relation.
Also, we need to find the set of all elements related to 1.
Since the relation is given by, , and 1 is an element of A,
.
Thus, the set of all element related to 1 is 1.
Question 9: Let be the set of all lines in XY-plane and
be the relation in
defined as
Show that
is an equivalence relation. Find the set of all lines related to the line
Answer:
To prove that relation is equivalence, we need to prove that it is reflexive, symmetric and transitive.
Reflexivity:
Since a line is always parallel to itself.
Therefore R is reflexive.
Symmetric:
Therefore R is symmetric.
Transitive:
Therefore R is transitive.
Since, R is reflexive, symmetric and transitive, so R is an equivalence relation.
And, the set of lines parallel to the line For all
where R is the set of real numbers.
Question 10: Show that the relation defined on the set
of all polygons as
is an equivalence relation. What is the set of all elements in A related to the right angle triangle T with sides 3, 4 and 5?
Answer:
Reflexivity:
R is reflexive since as the same polygon has the same number of sides with itself.
Symmetric:
Therefore R is symmetric.
Transitive:
Therefore R is transitive.
Hence, R is an equivalence relation. And, now the elements in A related to the right-angled triangle (T) with sides 3, 4 and 5 are those polygons which have three sides (since T is a polygon with three sides).
Hence, the set of all elements in A related to triangle T is the set of all triangles.
Question 11: Let be the origin. we define a relation between two points
and
in a plane if
Show that the relation, so defined is an equivalence relation.
Answer:
Let A be set of points on the plane. Let be a relation on A where O is the origin.
Reflexivity:
Therefore R is reflexive.
Symmetric:
Therefore R is symmetric.
Transitive:
Therefore R is transitive.
Since, R is reflexive, symmetric and transitive, so R is an equivalence relation on A.
Question 12: Let be the relation defined on the set
by
Show that
is an equivalence relation. Further, show that all the elements of the subset
are related to each other and all the elements of the subset
are related to each other, but no element of the subset
is related to any element of the subset
Answer:
Therefore,
Reflexivity :
From the relation R it is seen that R is reflexive.
Symmetric:
From the relation R, it is seen that R is symmetric.
Transitive:
From the relation R, it is seen that R is transitive too.
Also, from the relation R, it is seen that are related with each other only and
are related with each other.
Question 13: Let be a relation on the set
of all real numbers defined by
. Prove that
is not an equivalence relation on
.
Answer:
As this relation is not reflexive so it can’t be an equivalence relation.
Question 14: Let be the set of all integers and
be the set of all non-zero integers. Let a relation
on
be defined as follows:
Prove that is an equivalence relation on
Answer:
We have, be set of integers and
be the set of non-zero integers.
.
Reflexivity:
Therefore R is reflexive.
Symmetric:
Therefore R is symmetric.
Transitive:
Therefore R is transitive.
Hence, R is an equivalence relation on
Question 15: If and
are relations on a set
, then prove the following:
(i) and
are symmetric
and
are symmetric
(ii) is reflexive and
is any relation
is reflexive.
Answer:
R and S are two symmetric relations on set A
(i) To prove: is symmetric
Symmetric:
To prove:
Symmetric:
(ii) R and S are two relations on a such that R is reflexive.
To prove : is reflexive
Reflexivity:
Hence, is reflexive.
Question 16: If R and S are transitive relations on a set A, then prove that may not be a transitive relation on A.
Answer:
We will prove this using an example.
Let be a set and
and
are two relations on A
Clearly R and S are transitive relation on A
Now,
Here,
Therefore is not transitive
Question 17: Let be the set of all complex numbers and
be the set of all non-zero complex numbers. Let a relation
on
be defined as
Show that is an equivalence relation.
Answer:
Given: Set set of the non-zero complex number and a relation R in
is defined as:
To prove that R is equivalence relation, we have to prove that R is reflexive, symmetric and transitive.
(i) Reflexivity:
Therefore R is reflexive.
(ii) Symmetricity:
Therefore R is a symmetric relation.
(iii) Transitivity:
From equation i) and ii) we get
Therefore R is transitive relation