####
Chapter 1 - Relations And Functions
NCERT Solutions for Class 12 Science Math
Exercise No. 1.1

#### Page No 5:

##
Question 1:**Determine whether each of the following relations are reflexive, symmetric and transitive:**

**(i)Relation R in the set**

*A*= {1, 2, 3…13, 14} defined as**R = {(**

*x*,*y*): 3*x*−*y*= 0}

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

**R = {(**

*x*,*y*):*y*=*x*+ 5 and*x*< 4}

**(iii) Relation R in the set**

*A*= {1, 2, 3, 4, 5, 6} as**R = {(**

*x*,*y*):*y*is divisible by*x*}

**(iv) Relation R in the set Z of all integers defined as**

**R = {(**

*x*,*y*):*x*−*y*is as integer}

**(v) Relation R in the set**

*A*of human beings in a town at a particular time given by

**(a) R = {(**

*x*,*y*):*x*and*y*work at the same place}**(b) R = {(**

*x*,*y*):*x*and*y*live in the same locality}**(c) R = {(**

*x*,*y*):*x*is exactly 7 cm taller than*y*}**(d) R = {(**

*x*,*y*):*x*is wife of*y*}**(e) R = {(**

*x*,*y*):*x*is father of*y*}#### ANSWER:

##
**(i) ***A* = {1, 2, 3 … 13, 14}

*A*= {1, 2, 3 … 13, 14}

*x*,

*y*): 3

*x*−

*y*= 0}

##
**(ii) R = {(***x*, *y*): *y* = *x* + 5 and *x* < 4} = {(1, 6), (2, 7), (3, 8)}

*x*,

*y*):

*y*=

*x*+ 5 and

*x*< 4} = {(1, 6), (2, 7), (3, 8)}

*x*,

*y*) and (

*y*,

*z*) ∈R, then (

*x*,

*z*) cannot belong to R.

##
(iii) *A* = {1, 2, 3, 4, 5, 6}

*x*,

*y*):

*y*is divisible by

*x*}

*x)*is divisible by itself.

*x*,

*x*) ∈R

*x*,

*y*), (

*y*,

*z*) ∈ R. Then,

*y*is divisible by

*x*and

*z*is divisible by

*y*.

*z*is divisible by

*x*.

*x*,

*z*) ∈R

##
(iv) R = {(*x*, *y*): *x* − *y* is an integer}

*x*∈ Z, (

*x*,

*x*) ∈R as

*x*−

*x*= 0 is an integer.

*x*,

*y*∈ Z if (

*x*,

*y*) ∈ R, then

*x*−

*y*is an integer.

*x*−

*y*) is also an integer.

*y*−

*x*) is an integer.

*y*,

*x*) ∈ R

*x*,

*y*) and (

*y*,

*z*) ∈R, where

*x*,

*y*,

*z*∈ Z.

*x*−

*y*) and (

*y*−

*z*) are integers.

*x*−

*z*= (

*x*−

*y*) + (

*y*−

*z*) is an integer.

*x*,

*z*) ∈R

##

(v) (a) R = {(*x*, *y*): *x* and *y* work at the same place}

*x*,

*x*) ∈ R

*x*,

*y*) ∈ R, then

*x*and

*y*work at the same place.

*y*and

*x*work at the same place.

*y*,

*x*) ∈ R.

*x*,

*y*), (

*y*,

*z*) ∈ R

*x*and

*y*work at the same place and

*y*and

*z*work at the same place.

*x*and

*z*work at the same place.

*x*,

*z*) ∈R

##
**(b) R = {(***x*, *y*): *x* and *y* live in the same locality}

*x*,

*y*):

*x*and

*y*live in the same locality}

*x*,

*x*) ∈ R as

*x*and

*x*is the same human being.

*x*,

*y*) ∈R, then

*x*and

*y*live in the same locality.

*y*and

*x*live in the same locality.

*y*,

*x*) ∈ R

*x*,

*y*) ∈ R and (

*y*,

*z*) ∈ R.

*x*and

*y*live in the same locality and

*y*and

*z*live in the same locality.

*x*and

*z*live in the same locality.

*x,*

*z*) ∈ R

##
(c) R = {(*x*, *y*): *x* is exactly 7 cm taller than *y*}

*x*,

*x*) ∉ R

*x*cannot be taller than himself.

*x*,

*y*) ∈R.

*x*is exactly 7 cm taller than

*y*.

*y*is not taller than

*x*.

*y*,

*x*) ∉R

*x*is exactly 7 cm taller than

*y*, then

*y*is exactly 7 cm shorter than

*x*.

*x*,

*y*), (

*y*,

*z*) ∈ R.

*x*is exactly 7 cm taller than

*y*and

*y*is exactly 7 cm taller than z.

*x*is exactly 14 cm taller than

*z*.

*x*,

*z*) ∉R

##
**(d) R = {(***x*, *y*): *x* is the wife of *y*}

*x*,

*y*):

*x*is the wife of

*y*}

*x*,

*x*) ∉ R

*x*cannot be the wife of herself.

*x*,

*y*) ∈ R

*x*is the wife of

*y.*

*y*is not the wife of

*x*.

*y*,

*x*) ∉ R

*x*is the wife of

*y*, then

*y*is the husband of

*x*.

*x*,

*y*), (

*y*,

*z*) ∈ R

*x*is the wife of

*y*and

*y*is the wife of

*z*.

*x*is the wife of

*z*.

*x*,

*z*) ∉ R

##
(e) R = {(*x*, *y*): *x* is the father of *y*}

*x*,

*x*) ∉ R

*x*cannot be the father of himself.

*x*,

*y*) ∈R.

*x*is the father of

*y.*

*y*cannot be the father of

*y.*

*y*is the son or the daughter of

*y.*

*y*,

*x*) ∉ R

*x*,

*y*) ∈ R and (

*y*,

*z*) ∈ R.

*x*is the father of

*y*and

*y*is the father of

*z*.

*x*is not the father of

*z*.

*x*is the grandfather of

*z*.

*x*,

*z*) ∉ R

#### Page No 5:

##
Question 2:**Show that the relation R in the set R of real numbers, defined as ****R = {(***a*, *b*): *a* ≤ *b*2} is neither reflexive nor symmetric nor transitive.

*a*,

*b*):

*a*≤

*b*2} is neither reflexive nor symmetric nor transitive.

#### ANSWER:

*a*,

*b*):

*a*≤

*b*2}

#### Page No 5:

##
Question 3:**Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as****R = {(***a*, *b*): *b* = *a* + 1} is reflexive, symmetric or transitive.

*a*,

*b*):

*b*=

*a*+ 1} is reflexive, symmetric or transitive.

#### ANSWER:

*A*= {1, 2, 3, 4, 5, 6}.

*A*as:

*a*,

*b*):

*b*=

*a*+ 1}

*a*,

*a*) ∉ R, where

*a*∈ A.

#### Page No 5:

##
Question 4:Show that the relation R in R defined as R = {(*a*, *b*): *a* ≤ *b*}, is reflexive and transitive but not symmetric.

#### ANSWER:

*a*,

*b*);

*a*≤

*b*}

*a*,

*a*) ∈ R as

*a*=

*a*.

*a*,

*b*), (

*b*,

*c*) ∈ R.

*a*≤

*b*and

*b*≤

*c*

*a*≤

*c*

*a*,

*c*) ∈ R

#### Page No 5:

##
Question 5:Check whether the relation R in R defined as R = {(*a*, *b*): *a* ≤ *b*3} is reflexive, symmetric or transitive.

#### ANSWER:

*a*,

*b*):

*a*≤

*b*3}

#### Page No 6:

## Question 6:Show that the relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} is symmetric but neither reflexive nor transitive.

#### ANSWER:

*A*= {1, 2, 3}.

*A*is defined as R = {(1, 2), (2, 1)}.

#### Page No 6:

##
Question 7:Show that the relation R in the set *A* of all the books in a library of a college, given by R = {(*x*, *y*): *x* and *y* have same number of pages} is an equivalence relation.

#### ANSWER:

*A*is the set of all books in the library of a college.

*x*,

*y*):

*x*and

*y*have the same number of pages}

*x*,

*x*) ∈ R as

*x*and

*x*has the same number of pages.

*x*,

*y*) ∈ R ⇒

*x*and

*y*have the same number of pages.

*y*and

*x*have the same number of pages.

*y*,

*x*) ∈ R

*x*,

*y*) ∈R and (

*y*,

*z*) ∈ R.

*x*and

*y*and have the same number of pages and

*y*and

*z*have the same number of pages.

*x*and

*z*have the same number of pages.

*x*,

*z*) ∈ R

#### Page No 6:

##
Question 8:Show that the relation R in the set *A* = {1, 2, 3, 4, 5} given by, is an equivalence relation. Show that all the elements of {1, 3, 5} are related to each other and all the elements of {2, 4} are related to each other. But no element of {1, 3, 5} is related to any element of {2, 4}.

#### ANSWER:

*A*= {1, 2, 3, 4, 5}

*a*∈

*A*, we have (which is even).

*a*,

*b*) ∈ R.

*a*,

*b*) ∈ R and (

*b*,

*c*) ∈ R.

*a*,

*c*) ∈ R

#### Page No 6:

## Question 9:Show that each of the relation R in the set, given by (i) (ii) is an equivalence relation. Find the set of all elements related to 1 in each case.

#### ANSWER:

*a*∈A, we have (

*a*,

*a*) ∈ R as is a multiple of 4.

*a*,

*b*) ∈ R ⇒ is a multiple of 4.

*b*,

*a*) ∈ R

*a*,

*b*), (

*b*,

*c*) ∈ R.

*a*,

*c*) ∈R

*a*,

*b*):

*a*=

*b*}

*a*∈A, we have (

*a*,

*a*) ∈ R, since

*a*=

*a*.

*a*,

*b*) ∈ R.

*a*=

*b*

*b*=

*a*

*b*,

*a*) ∈ R

*a*,

*b*) ∈ R and (

*b*,

*c*) ∈ R.

*a*=

*b*and

*b*=

*c*

*a*=

*c*

*a*,

*c*) ∈ R

#### Page No 6:

## Question 10:Given an example of a relation. Which is(i) Symmetric but neither reflexive nor transitive.(ii) Transitive but neither reflexive nor symmetric.(iii) Reflexive and symmetric but not transitive.(iv) Reflexive and transitive but not symmetric.(v) Symmetric and transitive but not reflexive.

#### ANSWER:

*A*= {5, 6, 7}.

*A*as R = {(5, 6), (6, 5)}.

*a*,

*b*):

*a*<

*b*}

*a*∈ R, we have (

*a*,

*a*) ∉ R since

*a*cannot be strictly less than

*a*itself. In fact,

*a*=

*a*.

*a*,

*b*), (

*b*,

*c*) ∈ R.

*a*<

*b*and

*b*<

*c*

*a*<

*c*

*a*,

*c*) ∈ R

*A*= {4, 6, 8}.

*A*= {(4, 4), (6, 6), (8, 8), (4, 6), (6, 4), (6, 8), (8, 6)}

*a*∈

*A*, (

*a*,

*a*) ∈R i.e., (4, 4), (6, 6), (8, 8)} ∈ R.

*a*,

*b*) ∈ R ⇒ (

*b*,

*a*) ∈ R for all

*a*,

*b*∈ R.

*a*,

*b*):

*a*3 ≥

*b*3}

*a*,

*a*) ∈ R as

*a*3 =

*a*3.

*a*,

*b*), (

*b*,

*c*) ∈ R.

*a*3 ≥

*b*3 and

*b*3 ≥

*c*3

*a*3 ≥

*c*3

*a*,

*c*) ∈ R

*A*= {−5, −6}.

*A*as:

#### Page No 6:

##
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 point related to a point P ≠ (0, 0) is the circle passing through P with origin as centre.

#### ANSWER:

*k*, then the set of all points related to P is at a distance of

*k*from the origin.

#### Page No 6:

##
Question 12:Show that the relation R defined in the set *A* of all triangles as R = {(*T*1, *T*2): *T*1 is similar to *T*2}, is equivalence relation. Consider three right angle triangles *T*1 with sides 3, 4, 5, *T*2 with sides 5, 12, 13 and *T*3 with sides 6, 8, 10. Which triangles among *T*1, *T*2 and *T*3 are related?

#### ANSWER:

*T*1,

*T*2):

*T*1 is similar to

*T*2}

*T*1,

*T*2) ∈ R, then

*T*1 is similar to

*T*2.

*T*2 is similar to

*T*1.

*T*2,

*T*1) ∈R

*T*1,

*T*2), (

*T*2,

*T*3) ∈ R.

*T*1 is similar to

*T*2 and

*T*2 is similar to

*T*3.

*T*1 is similar to

*T*3.

*T*1,

*T*3) ∈ R

*T*1 and

*T*3 are in the same ratio.

*T*1 is similar to triangle

*T*3.

*T*1 is related to

*T*3.

#### Page No 6:

##
Question 13:Show that the relation R defined in the set *A* of all polygons as R = {(*P*1, *P*2): *P*1 and *P*2 have same number of sides}, 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:

*P*1,

*P*2):

*P*1 and

*P*2 have same the number of sides}

*P*1,

*P*1) ∈ R as the same polygon has the same number of sides with itself.

*P*1,

*P*2) ∈ R.

*P*1 and

*P*2 have the same number of sides.

*P*2 and

*P*1 have the same number of sides.

*P*2,

*P*1) ∈ R

*P*1,

*P*2), (

*P*2,

*P*3) ∈ R.

*P*1 and

*P*2 have the same number of sides. Also,

*P*2 and

*P*3 have the same number of sides.

*P*1 and

*P*3 have the same number of sides.

*P*1,

*P*3) ∈ R

*A*related to the right-angled triangle (

*T)*with sides 3, 4, and 5 are those polygons which have 3 sides (since

*T*is a polygon with 3 sides).

*A*related to triangle

*T*is the set of all triangles.

#### Page No 6:

##
Question 14:Let *L* be the set of all lines in XY plane and R be the relation in *L* defined as R = {(*L*1, *L**2*): *L**1* is parallel to *L*2}. Show that R is an equivalence relation. Find the set of all lines related to the line *y* = 2*x* + 4.

#### ANSWER:

*L*1,

*L*2): L1 is parallel to

*L*2}

*L*1 is parallel to itself i.e., (

*L*1,

*L*1) ∈ R.

*L*1,

*L*2) ∈ R.

*L*1 is parallel to

*L*2.

*L*2 is parallel to

*L*1.

*L*2,

*L*1) ∈ R

*L*1,

*L*2), (

*L*2,

*L*3) ∈R.

*L*1 is parallel to

*L*2. Also,

*L*2 is parallel to

*L*3.

*L*1 is parallel to

*L*3.

*y*= 2

*x*+ 4 is the set of all lines that are parallel to the line

*y*= 2

*x*+ 4.

*y*= 2

*x*+ 4 is

*m*= 2

*y*= 2

*x*+

*c*, where

*c*∈R.

*y*= 2

*x*+

*c*, where

*c*∈ R.

#### Page No 7:

## Question 15:Let R be the relation in the set {1, 2, 3, 4} given by R = {(1, 2), (2, 2), (1, 1), (4, 4), (1, 3), (3, 3), (3, 2)}. Choose the correct answer.(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.

#### ANSWER:

*a*,

*a*) ∈ R, for every

*a*∈{1, 2, 3, 4}.

*a*,

*b*), (

*b*,

*c*) ∈ R ⇒ (

*a*,

*c*) ∈ R for all

*a*,

*b*,

*c*∈ {1, 2, 3, 4}.

#### Page No 7:

##
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) (2, 4) ∈ R (B) (3, 8) ∈R (C) (6, 8) ∈R (D) (8, 7) ∈ R

#### ANSWER:

*a*,

*b*):

*a*=

*b*− 2,

*b*> 6}

*b*> 6, (2, 4) ∉ R