Exercise 1.1

Question 1: Determine whether each of the following relations are reflexive, symmetric and transitive.(Class 12 ncert solution math exercise 1.1)

(i) Relation in the set defined as

(ii) Relation in the set of natural numbers defined as

(iii) Relation in the set defined as

(iv) Relation in the set of integers defined as

(v) Relation in the set of human beings in a town at a particular time given by

(a) and work at the same place

(b) and live in the same locality

(c) is exactly taller than

(d) is wife of

(e) is father of

Solution:(i)

is not reflexive because and .

is not symmetric because , but since .

is not transitive because , but .

Hence, is neither reflexive nor symmetric nor transitive.

(ii)

is not reflexive because .

is not symmetric because but .

is not transitive because there isn’t any ordered pair in such that , so .

Hence, is neither reflexive nor symmetric nor transitive.

(iii) is divisible by

We know that any number other than 0 is divisible by itself.

Thus,

So, is reflexive. [because 4 is divisible by 2

But [since 2 is not divisible by 4

So, is not symmetric.

Let and . So, is divisible by and is divisible by .

So, is divisible by

So, is transitive.

So, is reflexive and transitive but not symmetric.

(iv) is an integer

For because is an integer.

So, is reflexive.

For, , if , then is an integer is an integer.

So,

So, is symmetric.

Let and , where .

and are integers.

is an integer.

So, is transitive.

So, is reflexive, symmetric and transitive.

(v)(a) and work at the same place

is reflexive because

is symmetric because,

If , then and work at the same place and and also work at the same place. .

is transitive because,

Let

and work at the same place and and work at the same place.

Then, and also works at the same place. .

Hence, is reflexive, symmetric and transitive.

(b) and live in the same locality

is reflexive because

is symmetric because,

If , then and live in the same locality and and also live in the same locality .

is transitive because, Let

and live in the same locality and and live in the same locality.

Then and also live in the same locality. .

Hence, is reflexive, symmetric and transitive.

(c) is exactly taller than

is not reflexive because .

is not symmetric because,

If , then is exactly taller than and is clearly not taller than .

is not transitive because,

Let

is exactly taller than and is exactly taller than .

Then is exactly taller than

Hence, is neither reflexive nor symmetric nor transitive.

(d) is wife of

is not reflexive because .

is not symmetric because,

Let is the wife of and is not the wife of .

is not transitive because,

Let

is wife of and is wife of , which is not possible.

.

Hence, is neither reflexive nor symmetric nor transitive.

(e) is father of

is not reflexive because .

is not symmetric because,

Let is the father of and is not the father of .

is not transitive because,

Let

is father of and is father of is not father of .

Hence, is neither reflexive nor symmetric nor transitive.

Question 2: Show that the relation R in the set of real numbers,defined as is neither reflexive nor symmetric nor transitive.

Solution:

because

is not reflexive.

as . But 4 is not less than .

is not symmetric.

[Because and

is not transitive.

is neither reflective nor symmetric nor transitive.

Question 3:  Check whether the relation defined in the set as is reflexive, symmetric or transitive.

Solution:

is not reflexive.

, but is not symmetric.

R is not transitive.

R is neither reflective nor symmetric nor transitive.

Question 4: Show that the relation in defined as is reflexive and transitive, but not symmetric.

Solution:

is reflexive.

as

as

is not symmetric.

and

is transitive.

is reflexive and transitive but not symmetric.

Question 5: Check whether the relation in defined as is reflexive, symmetric or transitive.

Solution:

Reflexive:- , since

is not reflexive.

Symmetric:-

R is not symmetric.

Transitive:-

is not transitive.

is neither reflexive nor symmetric nor transitive.

Question 6: Show that the relation in the set given by is symmetric but neither reflexive nor transitive.

Solution:

R is not reflexive.

and

R is symmetric.

and

R is not transitive.

is symmetric, but not reflexive or transitive.

Question 7: Show that the relation in the set of all books in a library of a college, given by and have same number of pages is an equivalence relation.

Solution: and have same number of pages

is reflexive since as and have same number of pages.  Therefore R is reflexive.

Since,

and have same number of pages and and have same number of pages

is symmetric.

and have same number of pages, and have same number of pages.

Then and have same number of pages.

is transitive.

is an equivalence relation.

Question 8: Show that the relation R in the set given by is even is an equivalence relation. 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 .

Solution:

(which is even)

is reflective.

is even

is transitive.

is an equivalence relation.

All elements of are related to each other because they are all odd. So, the modulus of the difference between any two elements is even.

Similarly, all elements are related to each other because they are all even.

No element of is related to any elements of as all elements of are odd and all elements of are even. So, the modulus of the difference between the two elements will not be even.

Question 9: Show that each of the relation R in the set , given by

(a). is a mutiple of 4

(b).

Is an equivalence relation. Find the set of all elements related to 1 in each case.

Solution:

(a) is a mutiple of 4

is a multiple of 4

is reflexive.

[is a multiple of 4 ]

[is a multiple of 4]

is symmetric.

and

is a multiple of 4 and is a multiple of 4

is a multiple of 4 and is a multiple of 4

is a multiple of 4

is a multiple of 4

is transitive.

is an equivalence relation.

The set of elements related to 1 is as

is a multiple of 4 .

is a multiple of 4 .

is a multiple of 4 .

(b)

[since ]

is reflective.

is symmetric.

and

and

is transitive.

is an equivalence relation.

The set of elements related to 1 is .

Question 10: Give an example of a relation, which is

(a) Symmetric but neither reflexive nor transitive.

(b) Transitive but neither reflexive nor symmetric.

(c)  Reflexive and symmetric but not transitive.

(d) Reflexive and transitive but not symmetric.

(e) Symmetric and transitive but not reflexive.

Solution:(a)

is not reflexive as

and is symmetric.

, but

is not transitive.

Relation is symmetric but not reflexive or transitive.

(b)

[since cannot be less than itself]

is not reflexive.

But 2 is not less than 1

is not symmetric.

is transitive.

Relation  R is transitive but not reflexive and symmetric.

(c)

is reflexive since

is symmetric since

for

is not transitive since , but

is reflexive and symmetric but not transitive.

(d)

is reflexive.

But is not symmetric.

is transitive.

is reflexive and transitive but not symmetric

(e) Let

is not reflexive as

is symmetric.

is transitive.

is symmetric and transitive but not reflexive.

Question 11: Show that the relation R in the set A of points in a plane given by R={(P, Q): Distance of the point P}from the origin is same as the distance of the point Q  from the origin } , is an equivalence relation. Further, show that the set of all points related to a point is the circle passing through P with origin as centre.

Solution: R={(P, Q) : Distance of the point P from the origin is same as the distance of the point Q from the origin}

Clearly,

is reflexive.

Clearly is symmetric.

⇒ The distance of P and Q from the origin is the same and also, the distance of Q and S from the origin is the same.

⇒  The distance of P and  S from the origin is the same.

is transitive. is an equivalence relation.

The set of points related to will be those points whose distance from origin is same as distance of from the origin.

Set of points forms a circle with the centre as origin and this circle passes through .

Question 12: Show that the relation R in the set  A of all triangles as is similar to , is an equivalence relation. Consider three right angle triangles with sides with sides and with sides . Which triangle among are related?

Solution: is similar to

R  is reflexive since every triangle is similar to itself.

If , then is similar to .

is similar to .

is symmetric.

is similar to and is similar to .

is similar to .

is transitive.

Corresponding sides of triangles and are in the same ratio.

Triangle is similar to triangle .

Hence, is related to .

Question 13: Show that the relation R in the set A of all polygons as and have same number of sides , is an equivalence relation. What is the set of all elements in A related to the right angle triangle with sides 3,4 and 5 ?

Solution: and have same number of sides

as same polygon has same number of sides.

is reflexive.

and have same number of sides.

and have same number of sides.

is symmetric.

and have same number of sides.

and have same number of sides.

and have same number of sides.

is transitive.

R is an equivalence relation.

The elements in A related to right-angled triangle (T) with sides are those polygons which have three sides.

Set of all elements in a related to triangle is the set of all triangles.

Question 14: Let L  be the set of all lines in plane and R be the relation in L defined as is parallel to . Show that R is an equivalence relation. Find the set of all lines related to the line .

Solution: is parallel to

is reflexive as any line is parallel to itself i.e.,

If , then

is parallel to .

is parallel to .

is symmetric.

is parallel to

is parallel to

is parallel to .

is transitive.

is an equivalence relation.

Set of all lines related to the line is the set of all lines that are parallel to the line .

Slope of the line is .

Line parallel to the given line is in the form , where .

Set of all lines related to the given line is given by , where .

Question 15: Let be the relation in the set given by

.

(a) R is reflexive and symmetric but not transitive.

(b) R is reflexive and transitive but not symmetric.

(c) R is symmetric and transitive but not reflexive.

(d) R  is an equivalence relation.

Solution:

for every

is reflexive.

but

is not symmetric.

for all

is not transitive.

is reflexive and transitive but not symmetric. The correct answer is B.

Question 16: Let  R be the relation in the set N given by R={(a, b): a=b-2, b>6}. Choose the correct answer.

(a)

(b)

(c)

(d)

Solution:

Now,

Consider

and