Question 1: Three relations are defined on set
as follows:
(i)
(ii)
(iii)
Find whether each one of is reflexive, symmetric and transitive.
Answer:
(i) Reflexive: Clearly, . Hence,
is reflexive on
.
Symmetric: We observe that is not a symmetric relation on
.
Transitive: We find that . Hence,
is not a transitive relation on
.
(ii) Reflexive: Since are not in
Hence, it is not a reflexive relation on
Symmetric: We find that the ordered pairs obtained by interchanging the components of ordered pairs in are also in
. Hence,
is a symmetric relation on
.
Transitive: Clearly,
Hence, it is not a transitive relation on .
(iii) Reflexive: Since none of is an element of
. Hence,
is not reflexive on
.
Symmetric: Clearly, . Hence,
is not a symmetric relation on
.
Transitive: Clearly, . Hence,
is not a transitive relation on
.
Question 2: Show that the relation on the set
given by
is reflexive but neither symmetric nor transitive.
Answer:
Since Hence,
is reflexive.
We observe that Hence,
is not symmetric.
Also, Hence,
is not transitive.
Question 3: Show that the relation on the set
given by
is symmetric but neither reflexive nor transitive.
Answer:
We observe that do not belong to
Hence,
is not reflexive.
Clearly, Hence,
is symmetric.
As Hence,
is not transitive
Question 4: Check the following relations for reflexivity, symmetry and transitivity:
Answer:
Symmetry: is not symmetric because if
then b may not divide
. For example,
but
Transitivity: Let such that
and
Then,
Hence, is a transitive relation on
(ii) Let be the set of ail lines in a plane. We are given that
Reflexivity: is not reflexive because a line cannot be perpendicular to itself i.e.
is not true.
Symmetry: Let Then,
Hence, is symmetric on
Transitive: is not transitive, because
and
does not imply that
Question 5: Let a relation on the set
of real numbers be defined as
for all
Show that
is reflexive and symmetric but not transitive.
Answer:
We observe the following properties:
Reflexivity: Let be an arbitrary element of
Then,
Question 6: Determine whether each of the following relations are reflexive, symmetric and transitive:
Answer:
Reflexivity: Clearly, etc. are not in
. Hence,
is not reflexive.
Symmetry: We find that but
Hence, R is not symmetric.
Transitivity: Since and there is no order pair in
which has
as the first element. Same is the case for
and
Hence,
is transitive.
Reflexivity: We know that is divisible by
for all
Symmetry: We observe that 6 is divisible by 2 but 2 is not divisible by 6. This means that
Hence, is not symmetric on set
Transitivity: Let
is divisible by
and
is divisible by
is divisible by
Hence, is transitive relation on
Reflexivity: We have, which is an integer for all
Question 7: Show that the relation defined as
, is reflexive and transitive but not symmetric.
Answer:
Question 8: Let be the set of all points in a plane and
be a relation on
defined as
Show that is reflexive and symmetric but not transitive.
Answer:
We observe the following properties of relation
Reflexivity: For any point in set
, we find that
Distance between and itself is 0 which is less than 2 units.
Hence, is reflexive on
Symmetry: Let and
be two points in
such that
Then,
Distance between
and
is less than 2 units.
Distance between
and
is less than 2 units
Hence, is symmetric on
Transitivity: Consider points having coordinates
We observe that the distance between
is 1.5 units which is less than 2 units and the distance between
is 1.7 units which is also less than 2 units. But, the distance between
is 3.2 which is not less than 2 units. This means that
Hence,
is not transitive on
Answer:
Clearly, are subsets of
In order to prove that
it is sufficient to show that
We observe that the difference between any two elements of each of the sets is a multiple of 3.
Let be an arbitrary element of
Then,
Now, let be an arbitrary element of
Then,
Therefore for (i) and (ii) we get
Question 10: Show that the relation R on the set R of all real numbers, defined as is neither reflexive nor symmetric nor transitive.
Answer:
Therefore, is not reflexive.
Hence, is not symmetric.
Transitivity: We observe that
Hence, is not transitive.
Question 11: Let Then, show that the number of relations containing
and
which are reflexive and transitive but not symmetric is three.
Answer:
The smallest reflexive relation on set containing
is
Since Hence,
is not transitive.
To make it transitive we have to include in
Including
in
we get
This is reflexive and transitive but not symmetric as but
Now, if we add the pair to
to get
The relation is reflexive and transitive but not symmetric. Similarly, by adding
and
respectively to
we get
These relations are reflexive and transitive but not symmetric.
We observe that out of ordered pairs and
at a time if we add any two ordered pairs at a time to
then to maintain the transitivity we will be forced to add the remaining third pair and in this process the relation will become symmetric also which is not required. Hence, the total number of reflexive, transitive but not symmetric relations containing
and
is three.
Question 12: Let be a relation on the set of all lines in a plane defined by
. Show that
is an equivalence relation.
Answer:
Let be the given set of lines in a plane. Then, we observe the following properties.
Reflexive: For each line , we have
is reflexive
Hence, R is transitive on L.
Hence, R being Reflexive, symmetric and transitive is an equivalence relation on L.
Question 13: Show that the relation is congruent to on the set of all triangles in a plane is an equivalence relation.
Answer:
Let be the set of all triangles in a plane and let
be the relation on
defined by
We observe the following properties of relation
Hence, R is symmetric on S
Hence, R is transitive on S. Hence, R being reflexive, symmetric and transitive, is an equivalence relation on S
Question 14: Show that the relation defined on the set
of all triangles in a plane as
is an equivalence relation. Consider three right angle triangles
with sides
with sides
and
with sides
. Which triangles among
are relate?
Answer:
We observe the following properties of relation R.
Reflexivity: We know that every triangle is similar to itself.
Hence, R is symmetric.
Hence, R is transitive.
Hence, R is an equivalence relation on set A.
In triangles and
, we observe that the corresponding angles are equal and the corresponding sides are proportional
Question 15: Let n be a positive integer. Prove that the relation R on the set Z of all integers numbers defined by is divisible by
, is an equivalence relation on Z.
Answer:
We observe the following properties of relation R.
Reflexivity: For any
Hence, R is transitive relation on Z.
Thus, R being reflexive, symmetric and transitive, is an equivalence relation on Z.
Question 16: Show that the relation R on the set A of all the books in a library of a college given by
is an equivalence relation.
Answer:
We observe the following properties of relation R
Reflexivity: For any book in set
, we observe that
and
have the same number of pages.
Hence, is reflexive.
and
have the same number of pages
and
have the same number of pages
Hence,
is symmetric.
Transitivity: Let . Then,
have the same number of pages) and (
have the same number of pages)
have the same number of pages.
Question 17: Show that the relation on the set
given by
Show that all the elements of
are related to each other and all the elements of
are related to each other. But, no element of
is related to any element of
Answer:
We have,
We observe the following properties of relation R.
Hence, R is reflexive.
Hence, R is symmetric.
Now two cases arise:
Case 1:
When is even in this case,
Case 2:
When is odd in this case,
Therefore,
Hence, R is an equivalence relation.
We know that the difference of any two odd (even) natural numbers is always an even natural number. Therefore, all the elements of set are related to each other and all the elements of
are related to each other.
We know that the difference of an even natural number and an odd natural number is an odd natural number. Therefore, no element of is related to any element of
.
Question 18: Show that the relation R on the set is an equivalence relation. Find the set of all elements related to 1 i.e. equivalence class [1].
Answer:
We have,
We observe the following properties of relation R.
Reflexivity: For any , we have
Hence, R is reflexive.
Hence, R is symmetric.
Hence, R is transitive.
Hence, R is an equivalence relation.
Let be an element of
such that
Hence, the set of all elements of A which are related to 1 is
Question 19: Show that the relation R on the set A of points in a plane, given by
is an equivalence relation. Further shaw that the set of alt points related to a point
is the circle passing through P with origin as center.
Answer:
Let denote the origin in the given plane. Then,
We observe the following properties of relation
Reflexivity: For any point in set
, we have
Hence, R is reflexive.
Symmetry: Let and
be two points in set
such that
Hence, R is symmetric.
Transitivity: Let P , Q and S be three points in set A such that
Hence, R is transitive.
Hence, R is an equivalence relation.
Lei P be a fixed point in set A and Q be any point in set A such that
moves in the plane in such a way that its distance from the origin
is always same and is equal to OP.
Locus of Q is a circle with center at the origin and radius OP.
Hence, the set of all points related to P is the circle passing through F with origin O as center.
Question 20: Prove that the relation R on the set defined by
is an equivalence relation.
Answer:
We observe the following properties of relation R.
Reflexivity: Let be an arbitrary element of
. Then,
Hence, R is symmetric on
Hence, R is transitive on
Hence, R being reflexive, symmetric and transitive, is an equivalence relation on .
Question 21: Let and R be the relation on
defined by
for all
. Prove that R is an equivalence relation and also obtain the equivalence class
.
Answer:
We observe the following properties of relation R.
Reflexivity: Let be an arbitrary element of
. Then,
Thus, for all
. Hence, R is reflexive on
Symmetry: Let be such that
. Then,
Hence, R is symmetric on
Hence, R is a transitive relation on
Hence, R is an equivalence relation on
Now,
Question 22: Let N be the set of all natural numbers and let R be a relation on defined by
Shorn that R is an equivalence relation on . Also, find the equivalence class
Answer:
We observe the following properties of relation R.
Reflexivity: Let be an arbitrary element of
. Then
Hence, R is reflexive on
Hence, R is symmetric on
Hence, R is transitive on
Hence, R being reflexive, symmetric and transitive, is an equivalence relation on
Question 23: Let N denote the set of all natural numbers and R be the relation on defined by
Check whether R is an equivalence relation on
Answer:
We observe the following properties of relation R.
Reflexivity: Let be an arbitrary element of
. Then,
Hence, R is reflexive on .
Hence, R is symmetric on .
and,
Adding i) and ii) we get
Hence, R is transitive on
Question 24: Prove that the relation ‘congruence modulo ‘ on the set Z of all integers is an equivalence relation.
Answer:
We observe the following properties of the given relation.
Let a be an arbitrary integer. Then,
So, “congruence modulo m” is reflexive.
Symmetry: Let a,b in Z such that a =b (mod m). Then,
So, “congruence modulo m” is symmetric on Z.
Question 25: Show that the number of equivalence relations on the set containing
latex (2, 1) is two.
Answer:
The smallest equivalence relation containing
and
is
Now, we are left with four ordered pairs namely and
. If we add any one, say
to
, then for symmetry we must add
and then for transitivity we are forced to add
and
. Thus, the only equivalence relation other than
is the universal relation. Hence, the total number of equivalence relations containing
and
is two.
Question 26: Given a non-empty set , consider
which is the set of all subsets of
. Define a relation in
as follows:
Is
an equivalence relation on
Justify your answer.
Answer:
Question 27: Let be the equivalence relation in the set
given by
divides
Write the equivalence class
Answer:
Clearly, the equivalence class is the set of those elements in
which are related to
under the relation
Question 28: On the set N of all natural numbers, a relation R is defined as follows:
Each of the natural numbers
and
leaves the same remainder less than 5 when divided by 5.
Show that R is an equivalence relation. Also, obtain the pairwise disjoint subsets determined by R.
Answer:
We observe the following properties of relation R.
Reflexivity: Let a be an arbitrary element of N. Then, either is less than 5 and if
, then on dividing
by 5 we obtain a remainder as one of the numbers
.
Thus, for all
. So, R is reflexive on N.
Symmetry: Let such that
. Then,
Each of
and
leaves the same remainder less than 5 when divided by 5
Each of
and
leave the same remainder less than 5 when divided by 5
Thus, for all
. So, R is symmetric.
Transitivity : Let be such that
and
Then,
Each of
and
leaves the same remainder less than 5 when divided by 5
Each of
and
leaves the same remainder less than 5 when divided by 5
Therefore Each of and
leaves the same remainder less than 5 when divided by 5
Thus, and
for all
So, R is a transitive relation on N.
Hence, R is an equivalence relation on N.
Let us now find the equivalence classes.
Proceeding in this manner we find that
and
Thus, we obtain the following disjoint equivalence classes: