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