Question 1: Let A be the set of all human beings in a town at a particular time. Determine whether each of the following relations are reflexive, symmetric and transitive:
Answer:
Reflexive:
Let be an arbitrary element of
. Then,
and
work at the same place
Thus, is reflexive relation.
Symmetric:
Let and
be an arbitrary element of
. Then,
and
work at the same place
and
work at the same place
Thus, is a symmetric relation.
Transitive:
Let and
be an arbitrary element of
. Then,
and
and
work at the same place
and
work at the same place
work at the same place
and
work at the same place
Thus, is a transitive relation.
Reflexive:
Let be an arbitrary element of
. Then,
and
live in the same locality
Thus, is a reflexive relation.
Symmetric:
Let and
be an arbitrary element of
. Then,
and
live in the same locality
and
live in the same locality
Thus, is a symmetric relation.
Transitive:
Let and
be an arbitrary element of
. Then,
and
and
live in the same locality
and
live in the same locality
live in the same locality
and
live in the same locality
Thus, is a transitive relation.
Reflexive:
Let be an arbitrary element of
. Let,
is wife of
is not possible
Thus, is not a reflexive relation.
Symmetric:
Let and
be an arbitrary element of
. Let,
is wife of
is husband of
Thus, is not a symmetric relation.
Transitive:
Let and
be an arbitrary element of
. Let,
and
is wife of
is not husband of
Thus, is not a transitive relation.
Reflexive:
Let be an arbitrary element of
. Let,
if father of
is not possible
Thus, is not a reflexive relation.
Symmetric:
Let and
be an arbitrary element of
. Let,
is father of
cannot be father of
Thus, is not a symmetric relation.
Transitive:
Let and
be an arbitrary element of
. Let,
and
is father of
is father of
Thus, is not a transitive relation.
Answer:
Answer:
Reflexivity:
Let be an arbitrary element of
.
Then,
Clearly, is not reflexive.
Symmetry:
Let
Then,
Hence, is symmetric.
Transitivity:
Here,
So, is not transitive.
Reflexivity:
Let be an arbitrary element of
.
Then,
On applying the given condition we get,
So, is reflexive.
Symmetry:
Let
So, is symmetric.
Transitivity:
Let
Clearly, is not transitive.
Reflexivity:
Let be an arbitrary element of
.
Then,
Clearly, is reflexive
Symmetry:
Let
But
So, is not symmetric.
Transitivity:
Let
But
Clearly, is not transitive.
Answer:
Given that
Reflexivity:
Here,
Clearly, is reflexive.
Symmetry:
Here, , But
So, is not symmetric.
Transitivity:
Here, But
So, is not transitive.
Now we consider
Given that
Reflexivity:
Clearly,
So, is not reflexive.
Symmetry:
Here,
Clearly, is symmetric.
Transitivity:
Here,
But
So, is not transitive.
Consider as
Given that
Reflexivity:
Clearly,
So, is not reflexive.
Symmetry:
Here,
So, is not symmetric.
Transitivity:
Here,
Clearly, is transitive.
Question 5: The following relations are defined on the set of real numbers:
Find whether these relations are reflexive, symmetric or transitive.
Answer:
Reflexivity:
Let be an arbitrary element of
Then,
Thus, this relation is not reflexive.
Symmetry:
Let
Therefore, the given relation is not symmetric.
Transitivity:
Let and
Then, and
Adding the two, we get
Clearly, the given relation is transitive.
Reflexivity:
Let be an arbitrary element of
Then,
i.e. [Since, square of any number is positive]
So, the given relation is reflexive.
Symmetry:
Let
So, the given relation is symmetric.
Transitivity:
Let and
Thus, the given relation is not transitive.
Reflexivity:
Let be an arbitrary element of
Then,
Clearly, is not reflexive.
Symmetry:
Let
for all
Thus, is not symmetric.
Transitivity:
Let and
and
Multiplying the corresponding sides, we get
So, is transitive.
Question 6: Check whether the relation defined on the set
as
is reflexive, symmetric or transitive.
Answer:
Let
relation
is defined on set
as
We observe that where
For instance,
Therefore is not reflexive.
It is also observed that but
Therefore is not symmetric.
Not, but
Therefore is not transitive.
Hence, is neither reflexive, not symmetric, nor transitive.
Question 7: Check whether the relation on R defined by
is reflexive, symmetric or transitive.
Answer:
Therefore, is not reflexive.
Now,
But
Therefore R is not symmetric.
Therefore is not transitive.
Hence, R is neither reflexive, nor symmetric, nor transitive.
Question 8: ‘ Prove that every identity relation on a set is reflexive, but the converse is not necessarily true.
Answer:
Let us first understand what ‘Reflexive Relation’ is and what ‘Identity Relation’ is.
Reflexive Relation: A binary relation over a set
is reflexive if every element of
is related to itself. Formally, this may be written as
Identity Relation: Let be any set.
Then the relation on
is called the identity relation on
Thus, in an identity relation, every element is related to itself only.
Let be a set.
Let be a binary relation defined on
Let is the identity relation on
Hence, every identity relation on set is reflexive by definition.
Converse: Let is the set.
Let be a relation defined on
is reflexive as per definition.
But,
is not identity relation by definition.
Hence, proved that every identity relation on a set is reflexive, but the converse is not necessarily true.
Question 9: . If , define relations on A which have properties of being
(i) reflexive, transitive but not symmetric.
(ii) symmetric but neither reflexive nor transitive.
(iii) reflexive, symmetric and transitive.
Answer:
(i) The relation on having properties of being reflexive, transitive, but not symmetric is
Relation satisfies reflexivity and transitivity.
And
However,
(ii) The relation on having properties of being reflexive, transitive, but not symmetric is
Relation satisfies reflexivity and transitivity.
And
However,
(iii) The relation on having properties of being symmetric, reflexive and transitive is
So, the is an equivalence relation on
Question 10: Let R be a relation defined on the set of natural numbers N as
Find the domain and range of R. Also, verify whether R is (i) reflexive, (ii) symmetric (iii) transit
Answer:
We have,
Since , largest value that
can take corresponds to the smallest value that
can take.
Range of is
such that
Since
Since, is not reflexive.
Also, since is not symmetric.
Therefore is not transitive.
Question 11: Is it true that every relation which is symmetric and transitive is also reflexive? Give reasons.
Answer:
It is not true that every relation which is symmetric and transitive is also reflexive.
Take for example:
Take a set
And define a relation on
.
Symmetric relation:
, is symmetric on set
.
Transitive relation:
, is the simplest transitive relation on set
.
is symmetric as well as transitive relation.
But is not reflexive here.
If only had it been reflexive.
Thus, it is not true that every relation which is symmetric and transitive is also reflexive.
Question 12: An integer is said to be related to another integer
if
is a multiple of
. Check if the relation is symmetric, reflexive and transitive.
Answer:
Given,
Reflexivity:
Thus, is reflexive relation.
Symmetry:
Let
Thus is not symmetric.
Transitivity:
Let
Here
Thus, is transitive.
Question 13: Show that the relation on the set
of all real numbers is reflexive and transitive but not symmetric.
Answer:
Let be the set such that
Reflexivity:
Let be an arbitrary element of
Hence, is reflexive.
Symmetry:
Let
Hence,
Hence, is not symmetric.
Transitivity:
Hence, is transitive.
Question 14: Give an example of a relation which is
(i) reflexive and symmetric but not transitive.
(ii) reflexive and transitive but not symmetric.
(iii) symmetric and transitive but not reflexive.
(iv) symmetric but neither reflexive nor transitive.
(v) transitive but neither reflexive nor symmetric
Answer:
(i) reflexive and symmetric but not transitive.
Let
Let be a relation on
such that
Relation is reflexive if for every
Relation is symmetric if
Relation is transitive if
(ii) reflexive and transitive but not symmetric.
Let
Let be a relation on
such that
Relation is reflexive if for every
Relation is symmetric if
Relation is transitive if
(iii) symmetric and transitive but not reflexive.
Let
Let be a relation on
such that
Relation is reflexive if for every
Relation is symmetric if
Relation is transitive if
(iv) symmetric but neither reflexive nor transitive.
Let
Let be a relation on
such that
Relation is reflexive if for every
Relation is symmetric if
Relation is transitive if
(v) transitive but neither reflexive nor symmetric
Let
Let be a relation on
such that
Relation is reflexive if for every
Relation is symmetric if
Relation is transitive if
Question 15: Given the relation on the set
add a minimum number of ordered pairs so that the enlarged relation is symmetric, transitive and reflexive.
Answer:
To make R an equivalence relation, it should be:
(i) Reflexive: So three more ordered pairs should be added to
to make it reflexive.
(ii) Symmetric : As contains
so two more ordered pairs
should be added to make it symmetric.
(iii) Transitive: So to make
transitive
should be added to
. Also to maintain the symmetric property
should then be added to
So, is an equivalence relation. So minimum 7 ordered pairs are to be added.
Question 16: Let and
be a relation on
What minimum number of ordered pairs may be added to R so that it may become a transitive relation on
Answer:
We have the relation such that
is defined on set
For transitive relation:
Note in
So, add
Now, we can see that is transitive. Hence, the ordered pair to be added is
Question 17: Let and the relation
be defined on
as follows:
Then, write minimum number of ordered pairs to be added in
to make it reflexive and transitive
Answer:
We have the relation such that
is defined on set
To make as reflexive we should add
and
to
Also, to make as transitive we should add
to
Hence, the minimum number of ordered pairs to be added in is
.
Question 18: Each of the following defines a relation on N:
Determine which of the above relations are reflexive, symmetric and transitive.
Answer:
If , then
which is not true for any
Therefore is not reflexive.
which is not true.
Therefore is not symmetric.
Therefore is transitive.
Therefore is not reflexive.
Also, is symmetric since for all
Therefore is not transitive.
Therefore is reflexive.
Therefore is symmetric.
Now if is square of an integer and
is square of an integer
Therefore is transitive.
Therefore is not reflexive.
Therefore is not symmetric.
Therefore is not transitive.