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: