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.