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1 | 318-321 | Now the inflow of brine brings salt into the tank at the rate of 5 kg per minute
(as 25 Γ 200 g = 5 kg) and the outflow of brine takes salt out of the tank at the rate of
25 1000
40
y
y
ο£ο£¬ο£«
ο£Ά
ο£Έο£· =
kg per minute (as at time t, the salt in the tank is 1000
y
kg) Thus, the rate of change of salt with respect to t is given by
dy
dt = 5
βy40
(Why )
or
1
40
dy
y
dt +
= 5 (1)
Rationalised 2023-24
206
MATHEMATICS
This gives a mathematical model for the given problem |
1 | 319-322 | Thus, the rate of change of salt with respect to t is given by
dy
dt = 5
βy40
(Why )
or
1
40
dy
y
dt +
= 5 (1)
Rationalised 2023-24
206
MATHEMATICS
This gives a mathematical model for the given problem Step 3 Equation (1) is a linear equation and can be easily solved |
1 | 320-323 | )
or
1
40
dy
y
dt +
= 5 (1)
Rationalised 2023-24
206
MATHEMATICS
This gives a mathematical model for the given problem Step 3 Equation (1) is a linear equation and can be easily solved The solution of (1) is
given by
40
40
200
C
t
t
ye
e
=
+
or y (t) = 200 + C
40
t
e
β |
1 | 321-324 | (1)
Rationalised 2023-24
206
MATHEMATICS
This gives a mathematical model for the given problem Step 3 Equation (1) is a linear equation and can be easily solved The solution of (1) is
given by
40
40
200
C
t
t
ye
e
=
+
or y (t) = 200 + C
40
t
e
β (2)
where, c is the constant of integration |
1 | 322-325 | Step 3 Equation (1) is a linear equation and can be easily solved The solution of (1) is
given by
40
40
200
C
t
t
ye
e
=
+
or y (t) = 200 + C
40
t
e
β (2)
where, c is the constant of integration Note that when t = 0, y = 250 |
1 | 323-326 | The solution of (1) is
given by
40
40
200
C
t
t
ye
e
=
+
or y (t) = 200 + C
40
t
e
β (2)
where, c is the constant of integration Note that when t = 0, y = 250 Therefore, 250 = 200 + C
or
C = 50
Then (2) reduces to
y = 200 + 50
40
t
e
β |
1 | 324-327 | (2)
where, c is the constant of integration Note that when t = 0, y = 250 Therefore, 250 = 200 + C
or
C = 50
Then (2) reduces to
y = 200 + 50
40
t
e
β (3)
or
200
50
yβ
=
40
t
e
β
or
40
t
e
=
50
200
yβ
Therefore
t = 40
50
200
loge
ο£ο£¬ο£«y β
ο£Έο£·ο£Ά |
1 | 325-328 | Note that when t = 0, y = 250 Therefore, 250 = 200 + C
or
C = 50
Then (2) reduces to
y = 200 + 50
40
t
e
β (3)
or
200
50
yβ
=
40
t
e
β
or
40
t
e
=
50
200
yβ
Therefore
t = 40
50
200
loge
ο£ο£¬ο£«y β
ο£Έο£·ο£Ά (4)
Here, the equation (4) gives the time t at which the salt in tank is y kg |
1 | 326-329 | Therefore, 250 = 200 + C
or
C = 50
Then (2) reduces to
y = 200 + 50
40
t
e
β (3)
or
200
50
yβ
=
40
t
e
β
or
40
t
e
=
50
200
yβ
Therefore
t = 40
50
200
loge
ο£ο£¬ο£«y β
ο£Έο£·ο£Ά (4)
Here, the equation (4) gives the time t at which the salt in tank is y kg Step 4 Since
40
t
e
β
is always positive, from (3), we conclude that y > 200 at all times
Thus, the minimum amount of salt content in the tank is 200 kg |
1 | 327-330 | (3)
or
200
50
yβ
=
40
t
e
β
or
40
t
e
=
50
200
yβ
Therefore
t = 40
50
200
loge
ο£ο£¬ο£«y β
ο£Έο£·ο£Ά (4)
Here, the equation (4) gives the time t at which the salt in tank is y kg Step 4 Since
40
t
e
β
is always positive, from (3), we conclude that y > 200 at all times
Thus, the minimum amount of salt content in the tank is 200 kg Also, from (4), we conclude that t > 0 if and only if 0 < y β 200 < 50 i |
1 | 328-331 | (4)
Here, the equation (4) gives the time t at which the salt in tank is y kg Step 4 Since
40
t
e
β
is always positive, from (3), we conclude that y > 200 at all times
Thus, the minimum amount of salt content in the tank is 200 kg Also, from (4), we conclude that t > 0 if and only if 0 < y β 200 < 50 i e |
1 | 329-332 | Step 4 Since
40
t
e
β
is always positive, from (3), we conclude that y > 200 at all times
Thus, the minimum amount of salt content in the tank is 200 kg Also, from (4), we conclude that t > 0 if and only if 0 < y β 200 < 50 i e , if and only
if 200 < y < 250 i |
1 | 330-333 | Also, from (4), we conclude that t > 0 if and only if 0 < y β 200 < 50 i e , if and only
if 200 < y < 250 i e |
1 | 331-334 | e , if and only
if 200 < y < 250 i e , the amount of salt content in the tank after the start of inflow and
outflow of the brine is between 200 kg and 250 kg |
1 | 332-335 | , if and only
if 200 < y < 250 i e , the amount of salt content in the tank after the start of inflow and
outflow of the brine is between 200 kg and 250 kg Limitations of Mathematical Modelling
Till today many mathematical models have been developed and applied successfully
to understand and get an insight into thousands of situations |
1 | 333-336 | e , the amount of salt content in the tank after the start of inflow and
outflow of the brine is between 200 kg and 250 kg Limitations of Mathematical Modelling
Till today many mathematical models have been developed and applied successfully
to understand and get an insight into thousands of situations Some of the subjects like
mathematical physics, mathematical economics, operations research, bio-mathematics
etc |
1 | 334-337 | , the amount of salt content in the tank after the start of inflow and
outflow of the brine is between 200 kg and 250 kg Limitations of Mathematical Modelling
Till today many mathematical models have been developed and applied successfully
to understand and get an insight into thousands of situations Some of the subjects like
mathematical physics, mathematical economics, operations research, bio-mathematics
etc are almost synonymous with mathematical modelling |
1 | 335-338 | Limitations of Mathematical Modelling
Till today many mathematical models have been developed and applied successfully
to understand and get an insight into thousands of situations Some of the subjects like
mathematical physics, mathematical economics, operations research, bio-mathematics
etc are almost synonymous with mathematical modelling But there are still a large number of situations which are yet to be modelled |
1 | 336-339 | Some of the subjects like
mathematical physics, mathematical economics, operations research, bio-mathematics
etc are almost synonymous with mathematical modelling But there are still a large number of situations which are yet to be modelled The
reason behind this is that either the situation are found to be very complex or the
mathematical models formed are mathematically intractable |
1 | 337-340 | are almost synonymous with mathematical modelling But there are still a large number of situations which are yet to be modelled The
reason behind this is that either the situation are found to be very complex or the
mathematical models formed are mathematically intractable Rationalised 2023-24
MATHEMATICAL MODELLING 207
The development of the powerful computers and super computers has enabled us
to mathematically model a large number of situations (even complex situations) |
1 | 338-341 | But there are still a large number of situations which are yet to be modelled The
reason behind this is that either the situation are found to be very complex or the
mathematical models formed are mathematically intractable Rationalised 2023-24
MATHEMATICAL MODELLING 207
The development of the powerful computers and super computers has enabled us
to mathematically model a large number of situations (even complex situations) Due
to these fast and advanced computers, it has been possible to prepare more realistic
models which can obtain better agreements with observations |
1 | 339-342 | The
reason behind this is that either the situation are found to be very complex or the
mathematical models formed are mathematically intractable Rationalised 2023-24
MATHEMATICAL MODELLING 207
The development of the powerful computers and super computers has enabled us
to mathematically model a large number of situations (even complex situations) Due
to these fast and advanced computers, it has been possible to prepare more realistic
models which can obtain better agreements with observations However, we do not have good guidelines for choosing various parameters / variables
and also for estimating the values of these parameters / variables used in a mathematical
model |
1 | 340-343 | Rationalised 2023-24
MATHEMATICAL MODELLING 207
The development of the powerful computers and super computers has enabled us
to mathematically model a large number of situations (even complex situations) Due
to these fast and advanced computers, it has been possible to prepare more realistic
models which can obtain better agreements with observations However, we do not have good guidelines for choosing various parameters / variables
and also for estimating the values of these parameters / variables used in a mathematical
model Infact, we can prepare reasonably accurate models to fit any data by choosing
five or six parameters / variables |
1 | 341-344 | Due
to these fast and advanced computers, it has been possible to prepare more realistic
models which can obtain better agreements with observations However, we do not have good guidelines for choosing various parameters / variables
and also for estimating the values of these parameters / variables used in a mathematical
model Infact, we can prepare reasonably accurate models to fit any data by choosing
five or six parameters / variables We require a minimal number of parameters / variables
to be able to estimate them accurately |
1 | 342-345 | However, we do not have good guidelines for choosing various parameters / variables
and also for estimating the values of these parameters / variables used in a mathematical
model Infact, we can prepare reasonably accurate models to fit any data by choosing
five or six parameters / variables We require a minimal number of parameters / variables
to be able to estimate them accurately Mathematical modelling of large or complex situations has its own special problems |
1 | 343-346 | Infact, we can prepare reasonably accurate models to fit any data by choosing
five or six parameters / variables We require a minimal number of parameters / variables
to be able to estimate them accurately Mathematical modelling of large or complex situations has its own special problems These type of situations usually occur in the study of world models of environment,
oceanography, pollution control etc |
1 | 344-347 | We require a minimal number of parameters / variables
to be able to estimate them accurately Mathematical modelling of large or complex situations has its own special problems These type of situations usually occur in the study of world models of environment,
oceanography, pollution control etc Mathematical modellers from all disciplines β
mathematics, computer science, physics, engineering, social sciences, etc |
1 | 345-348 | Mathematical modelling of large or complex situations has its own special problems These type of situations usually occur in the study of world models of environment,
oceanography, pollution control etc Mathematical modellers from all disciplines β
mathematics, computer science, physics, engineering, social sciences, etc , are involved
in meeting these challenges with courage |
1 | 346-349 | These type of situations usually occur in the study of world models of environment,
oceanography, pollution control etc Mathematical modellers from all disciplines β
mathematics, computer science, physics, engineering, social sciences, etc , are involved
in meeting these challenges with courage βv
v
v
v
vβ
Rationalised 2023-24
vThere is no permanent place in the world for ugly mathematics |
1 | 347-350 | Mathematical modellers from all disciplines β
mathematics, computer science, physics, engineering, social sciences, etc , are involved
in meeting these challenges with courage βv
v
v
v
vβ
Rationalised 2023-24
vThere is no permanent place in the world for ugly mathematics It may
be very hard to define mathematical beauty but that is just as true of
beauty of any kind, we may not know quite what we mean by a
beautiful poem, but that does not prevent us from recognising
one when we read it |
1 | 348-351 | , are involved
in meeting these challenges with courage βv
v
v
v
vβ
Rationalised 2023-24
vThere is no permanent place in the world for ugly mathematics It may
be very hard to define mathematical beauty but that is just as true of
beauty of any kind, we may not know quite what we mean by a
beautiful poem, but that does not prevent us from recognising
one when we read it β G |
1 | 349-352 | βv
v
v
v
vβ
Rationalised 2023-24
vThere is no permanent place in the world for ugly mathematics It may
be very hard to define mathematical beauty but that is just as true of
beauty of any kind, we may not know quite what we mean by a
beautiful poem, but that does not prevent us from recognising
one when we read it β G H |
1 | 350-353 | It may
be very hard to define mathematical beauty but that is just as true of
beauty of any kind, we may not know quite what we mean by a
beautiful poem, but that does not prevent us from recognising
one when we read it β G H HARDY v
1 |
1 | 351-354 | β G H HARDY v
1 1 Introduction
Recall that the notion of relations and functions, domain,
co-domain and range have been introduced in Class XI
along with different types of specific real valued functions
and their graphs |
1 | 352-355 | H HARDY v
1 1 Introduction
Recall that the notion of relations and functions, domain,
co-domain and range have been introduced in Class XI
along with different types of specific real valued functions
and their graphs The concept of the term βrelationβ in
mathematics has been drawn from the meaning of relation
in English language, according to which two objects or
quantities are related if there is a recognisable connection
or link between the two objects or quantities |
1 | 353-356 | HARDY v
1 1 Introduction
Recall that the notion of relations and functions, domain,
co-domain and range have been introduced in Class XI
along with different types of specific real valued functions
and their graphs The concept of the term βrelationβ in
mathematics has been drawn from the meaning of relation
in English language, according to which two objects or
quantities are related if there is a recognisable connection
or link between the two objects or quantities Let A be
the set of students of Class XII of a school and B be the
set of students of Class XI of the same school |
1 | 354-357 | 1 Introduction
Recall that the notion of relations and functions, domain,
co-domain and range have been introduced in Class XI
along with different types of specific real valued functions
and their graphs The concept of the term βrelationβ in
mathematics has been drawn from the meaning of relation
in English language, according to which two objects or
quantities are related if there is a recognisable connection
or link between the two objects or quantities Let A be
the set of students of Class XII of a school and B be the
set of students of Class XI of the same school Then some
of the examples of relations from A to B are
(i)
{(a, b) β A Γ B: a is brother of b},
(ii)
{(a, b) β A Γ B: a is sister of b},
(iii)
{(a, b) β A Γ B: age of a is greater than age of b},
(iv)
{(a, b) β A Γ B: total marks obtained by a in the final examination is less than
the total marks obtained by b in the final examination},
(v)
{(a, b) β A Γ B: a lives in the same locality as b} |
1 | 355-358 | The concept of the term βrelationβ in
mathematics has been drawn from the meaning of relation
in English language, according to which two objects or
quantities are related if there is a recognisable connection
or link between the two objects or quantities Let A be
the set of students of Class XII of a school and B be the
set of students of Class XI of the same school Then some
of the examples of relations from A to B are
(i)
{(a, b) β A Γ B: a is brother of b},
(ii)
{(a, b) β A Γ B: a is sister of b},
(iii)
{(a, b) β A Γ B: age of a is greater than age of b},
(iv)
{(a, b) β A Γ B: total marks obtained by a in the final examination is less than
the total marks obtained by b in the final examination},
(v)
{(a, b) β A Γ B: a lives in the same locality as b} However, abstracting from
this, we define mathematically a relation R from A to B as an arbitrary subset
of A Γ B |
1 | 356-359 | Let A be
the set of students of Class XII of a school and B be the
set of students of Class XI of the same school Then some
of the examples of relations from A to B are
(i)
{(a, b) β A Γ B: a is brother of b},
(ii)
{(a, b) β A Γ B: a is sister of b},
(iii)
{(a, b) β A Γ B: age of a is greater than age of b},
(iv)
{(a, b) β A Γ B: total marks obtained by a in the final examination is less than
the total marks obtained by b in the final examination},
(v)
{(a, b) β A Γ B: a lives in the same locality as b} However, abstracting from
this, we define mathematically a relation R from A to B as an arbitrary subset
of A Γ B If (a, b) β R, we say that a is related to b under the relation R and we write as
a R b |
1 | 357-360 | Then some
of the examples of relations from A to B are
(i)
{(a, b) β A Γ B: a is brother of b},
(ii)
{(a, b) β A Γ B: a is sister of b},
(iii)
{(a, b) β A Γ B: age of a is greater than age of b},
(iv)
{(a, b) β A Γ B: total marks obtained by a in the final examination is less than
the total marks obtained by b in the final examination},
(v)
{(a, b) β A Γ B: a lives in the same locality as b} However, abstracting from
this, we define mathematically a relation R from A to B as an arbitrary subset
of A Γ B If (a, b) β R, we say that a is related to b under the relation R and we write as
a R b In general, (a, b) β R, we do not bother whether there is a recognisable
connection or link between a and b |
1 | 358-361 | However, abstracting from
this, we define mathematically a relation R from A to B as an arbitrary subset
of A Γ B If (a, b) β R, we say that a is related to b under the relation R and we write as
a R b In general, (a, b) β R, we do not bother whether there is a recognisable
connection or link between a and b As seen in Class XI, functions are special kind of
relations |
1 | 359-362 | If (a, b) β R, we say that a is related to b under the relation R and we write as
a R b In general, (a, b) β R, we do not bother whether there is a recognisable
connection or link between a and b As seen in Class XI, functions are special kind of
relations In this chapter, we will study different types of relations and functions, composition
of functions, invertible functions and binary operations |
1 | 360-363 | In general, (a, b) β R, we do not bother whether there is a recognisable
connection or link between a and b As seen in Class XI, functions are special kind of
relations In this chapter, we will study different types of relations and functions, composition
of functions, invertible functions and binary operations Chapter 1
RELATIONS AND FUNCTIONS
Lejeune Dirichlet
(1805-1859)
Rationalised 2023-24
MATHEMATICS
2
1 |
1 | 361-364 | As seen in Class XI, functions are special kind of
relations In this chapter, we will study different types of relations and functions, composition
of functions, invertible functions and binary operations Chapter 1
RELATIONS AND FUNCTIONS
Lejeune Dirichlet
(1805-1859)
Rationalised 2023-24
MATHEMATICS
2
1 2 Types of Relations
In this section, we would like to study different types of relations |
1 | 362-365 | In this chapter, we will study different types of relations and functions, composition
of functions, invertible functions and binary operations Chapter 1
RELATIONS AND FUNCTIONS
Lejeune Dirichlet
(1805-1859)
Rationalised 2023-24
MATHEMATICS
2
1 2 Types of Relations
In this section, we would like to study different types of relations We know that a
relation in a set A is a subset of A Γ A |
1 | 363-366 | Chapter 1
RELATIONS AND FUNCTIONS
Lejeune Dirichlet
(1805-1859)
Rationalised 2023-24
MATHEMATICS
2
1 2 Types of Relations
In this section, we would like to study different types of relations We know that a
relation in a set A is a subset of A Γ A Thus, the empty set Ο and A Γ A are two
extreme relations |
1 | 364-367 | 2 Types of Relations
In this section, we would like to study different types of relations We know that a
relation in a set A is a subset of A Γ A Thus, the empty set Ο and A Γ A are two
extreme relations For illustration, consider a relation R in the set A = {1, 2, 3, 4} given by
R = {(a, b): a β b = 10} |
1 | 365-368 | We know that a
relation in a set A is a subset of A Γ A Thus, the empty set Ο and A Γ A are two
extreme relations For illustration, consider a relation R in the set A = {1, 2, 3, 4} given by
R = {(a, b): a β b = 10} This is the empty set, as no pair (a, b) satisfies the condition
a β b = 10 |
1 | 366-369 | Thus, the empty set Ο and A Γ A are two
extreme relations For illustration, consider a relation R in the set A = {1, 2, 3, 4} given by
R = {(a, b): a β b = 10} This is the empty set, as no pair (a, b) satisfies the condition
a β b = 10 Similarly, Rβ² = {(a, b) : | a β b | β₯ 0} is the whole set A Γ A, as all pairs
(a, b) in A Γ A satisfy | a β b | β₯ 0 |
1 | 367-370 | For illustration, consider a relation R in the set A = {1, 2, 3, 4} given by
R = {(a, b): a β b = 10} This is the empty set, as no pair (a, b) satisfies the condition
a β b = 10 Similarly, Rβ² = {(a, b) : | a β b | β₯ 0} is the whole set A Γ A, as all pairs
(a, b) in A Γ A satisfy | a β b | β₯ 0 These two extreme examples lead us to the
following definitions |
1 | 368-371 | This is the empty set, as no pair (a, b) satisfies the condition
a β b = 10 Similarly, Rβ² = {(a, b) : | a β b | β₯ 0} is the whole set A Γ A, as all pairs
(a, b) in A Γ A satisfy | a β b | β₯ 0 These two extreme examples lead us to the
following definitions Definition 1 A relation R in a set A is called empty relation, if no element of A is
related to any element of A, i |
1 | 369-372 | Similarly, Rβ² = {(a, b) : | a β b | β₯ 0} is the whole set A Γ A, as all pairs
(a, b) in A Γ A satisfy | a β b | β₯ 0 These two extreme examples lead us to the
following definitions Definition 1 A relation R in a set A is called empty relation, if no element of A is
related to any element of A, i e |
1 | 370-373 | These two extreme examples lead us to the
following definitions Definition 1 A relation R in a set A is called empty relation, if no element of A is
related to any element of A, i e , R = Ο β A Γ A |
1 | 371-374 | Definition 1 A relation R in a set A is called empty relation, if no element of A is
related to any element of A, i e , R = Ο β A Γ A Definition 2 A relation R in a set A is called universal relation, if each element of A
is related to every element of A, i |
1 | 372-375 | e , R = Ο β A Γ A Definition 2 A relation R in a set A is called universal relation, if each element of A
is related to every element of A, i e |
1 | 373-376 | , R = Ο β A Γ A Definition 2 A relation R in a set A is called universal relation, if each element of A
is related to every element of A, i e , R = A Γ A |
1 | 374-377 | Definition 2 A relation R in a set A is called universal relation, if each element of A
is related to every element of A, i e , R = A Γ A Both the empty relation and the universal relation are some times called trivial
relations |
1 | 375-378 | e , R = A Γ A Both the empty relation and the universal relation are some times called trivial
relations Example 1 Let A be the set of all students of a boys school |
1 | 376-379 | , R = A Γ A Both the empty relation and the universal relation are some times called trivial
relations Example 1 Let A be the set of all students of a boys school Show that the relation R
in A given by R = {(a, b) : a is sister of b} is the empty relation and Rβ² = {(a, b) : the
difference between heights of a and b is less than 3 meters} is the universal relation |
1 | 377-380 | Both the empty relation and the universal relation are some times called trivial
relations Example 1 Let A be the set of all students of a boys school Show that the relation R
in A given by R = {(a, b) : a is sister of b} is the empty relation and Rβ² = {(a, b) : the
difference between heights of a and b is less than 3 meters} is the universal relation Solution Since the school is boys school, no student of the school can be sister of any
student of the school |
1 | 378-381 | Example 1 Let A be the set of all students of a boys school Show that the relation R
in A given by R = {(a, b) : a is sister of b} is the empty relation and Rβ² = {(a, b) : the
difference between heights of a and b is less than 3 meters} is the universal relation Solution Since the school is boys school, no student of the school can be sister of any
student of the school Hence, R = Ο, showing that R is the empty relation |
1 | 379-382 | Show that the relation R
in A given by R = {(a, b) : a is sister of b} is the empty relation and Rβ² = {(a, b) : the
difference between heights of a and b is less than 3 meters} is the universal relation Solution Since the school is boys school, no student of the school can be sister of any
student of the school Hence, R = Ο, showing that R is the empty relation It is also
obvious that the difference between heights of any two students of the school has to be
less than 3 meters |
1 | 380-383 | Solution Since the school is boys school, no student of the school can be sister of any
student of the school Hence, R = Ο, showing that R is the empty relation It is also
obvious that the difference between heights of any two students of the school has to be
less than 3 meters This shows that Rβ² = A Γ A is the universal relation |
1 | 381-384 | Hence, R = Ο, showing that R is the empty relation It is also
obvious that the difference between heights of any two students of the school has to be
less than 3 meters This shows that Rβ² = A Γ A is the universal relation Remark In Class XI, we have seen two ways of representing a relation, namely raster
method and set builder method |
1 | 382-385 | It is also
obvious that the difference between heights of any two students of the school has to be
less than 3 meters This shows that Rβ² = A Γ A is the universal relation Remark In Class XI, we have seen two ways of representing a relation, namely raster
method and set builder method However, a relation R in the set {1, 2, 3, 4} defined by R
= {(a, b) : b = a + 1} is also expressed as a R b if and only if
b = a + 1 by many authors |
1 | 383-386 | This shows that Rβ² = A Γ A is the universal relation Remark In Class XI, we have seen two ways of representing a relation, namely raster
method and set builder method However, a relation R in the set {1, 2, 3, 4} defined by R
= {(a, b) : b = a + 1} is also expressed as a R b if and only if
b = a + 1 by many authors We may also use this notation, as and when convenient |
1 | 384-387 | Remark In Class XI, we have seen two ways of representing a relation, namely raster
method and set builder method However, a relation R in the set {1, 2, 3, 4} defined by R
= {(a, b) : b = a + 1} is also expressed as a R b if and only if
b = a + 1 by many authors We may also use this notation, as and when convenient If (a, b) β R, we say that a is related to b and we denote it as a R b |
1 | 385-388 | However, a relation R in the set {1, 2, 3, 4} defined by R
= {(a, b) : b = a + 1} is also expressed as a R b if and only if
b = a + 1 by many authors We may also use this notation, as and when convenient If (a, b) β R, we say that a is related to b and we denote it as a R b One of the most important relation, which plays a significant role in Mathematics,
is an equivalence relation |
1 | 386-389 | We may also use this notation, as and when convenient If (a, b) β R, we say that a is related to b and we denote it as a R b One of the most important relation, which plays a significant role in Mathematics,
is an equivalence relation To study equivalence relation, we first consider three
types of relations, namely reflexive, symmetric and transitive |
1 | 387-390 | If (a, b) β R, we say that a is related to b and we denote it as a R b One of the most important relation, which plays a significant role in Mathematics,
is an equivalence relation To study equivalence relation, we first consider three
types of relations, namely reflexive, symmetric and transitive Definition 3 A relation R in a set A is called
(i)
reflexive, if (a, a) β R, for every a β A,
(ii)
symmetric, if (a1, a2) β R implies that (a2, a1) β R, for all a1, a2 β A |
1 | 388-391 | One of the most important relation, which plays a significant role in Mathematics,
is an equivalence relation To study equivalence relation, we first consider three
types of relations, namely reflexive, symmetric and transitive Definition 3 A relation R in a set A is called
(i)
reflexive, if (a, a) β R, for every a β A,
(ii)
symmetric, if (a1, a2) β R implies that (a2, a1) β R, for all a1, a2 β A (iii)
transitive, if (a1, a2) β R and (a2, a3) β R implies that (a1, a3) β R, for all a1, a2,
a3 β A |
1 | 389-392 | To study equivalence relation, we first consider three
types of relations, namely reflexive, symmetric and transitive Definition 3 A relation R in a set A is called
(i)
reflexive, if (a, a) β R, for every a β A,
(ii)
symmetric, if (a1, a2) β R implies that (a2, a1) β R, for all a1, a2 β A (iii)
transitive, if (a1, a2) β R and (a2, a3) β R implies that (a1, a3) β R, for all a1, a2,
a3 β A Rationalised 2023-24
RELATIONS AND FUNCTIONS
3
Definition 4 A relation R in a set A is said to be an equivalence relation if R is
reflexive, symmetric and transitive |
1 | 390-393 | Definition 3 A relation R in a set A is called
(i)
reflexive, if (a, a) β R, for every a β A,
(ii)
symmetric, if (a1, a2) β R implies that (a2, a1) β R, for all a1, a2 β A (iii)
transitive, if (a1, a2) β R and (a2, a3) β R implies that (a1, a3) β R, for all a1, a2,
a3 β A Rationalised 2023-24
RELATIONS AND FUNCTIONS
3
Definition 4 A relation R in a set A is said to be an equivalence relation if R is
reflexive, symmetric and transitive Example 2 Let T be the set of all triangles in a plane with R a relation in T given by
R = {(T1, T2) : T1 is congruent to T2} |
1 | 391-394 | (iii)
transitive, if (a1, a2) β R and (a2, a3) β R implies that (a1, a3) β R, for all a1, a2,
a3 β A Rationalised 2023-24
RELATIONS AND FUNCTIONS
3
Definition 4 A relation R in a set A is said to be an equivalence relation if R is
reflexive, symmetric and transitive Example 2 Let T be the set of all triangles in a plane with R a relation in T given by
R = {(T1, T2) : T1 is congruent to T2} Show that R is an equivalence relation |
1 | 392-395 | Rationalised 2023-24
RELATIONS AND FUNCTIONS
3
Definition 4 A relation R in a set A is said to be an equivalence relation if R is
reflexive, symmetric and transitive Example 2 Let T be the set of all triangles in a plane with R a relation in T given by
R = {(T1, T2) : T1 is congruent to T2} Show that R is an equivalence relation Solution R is reflexive, since every triangle is congruent to itself |
1 | 393-396 | Example 2 Let T be the set of all triangles in a plane with R a relation in T given by
R = {(T1, T2) : T1 is congruent to T2} Show that R is an equivalence relation Solution R is reflexive, since every triangle is congruent to itself Further,
(T1, T2) β R β T1 is congruent to T2 β T2 is congruent to T1 β (T2, T1) β R |
1 | 394-397 | Show that R is an equivalence relation Solution R is reflexive, since every triangle is congruent to itself Further,
(T1, T2) β R β T1 is congruent to T2 β T2 is congruent to T1 β (T2, T1) β R Hence,
R is symmetric |
1 | 395-398 | Solution R is reflexive, since every triangle is congruent to itself Further,
(T1, T2) β R β T1 is congruent to T2 β T2 is congruent to T1 β (T2, T1) β R Hence,
R is symmetric Moreover, (T1, T2), (T2, T3) β R β T1 is congruent to T2 and T2 is
congruent to T3 β T1 is congruent to T3 β (T1, T3) β R |
1 | 396-399 | Further,
(T1, T2) β R β T1 is congruent to T2 β T2 is congruent to T1 β (T2, T1) β R Hence,
R is symmetric Moreover, (T1, T2), (T2, T3) β R β T1 is congruent to T2 and T2 is
congruent to T3 β T1 is congruent to T3 β (T1, T3) β R Therefore, R is an equivalence
relation |
1 | 397-400 | Hence,
R is symmetric Moreover, (T1, T2), (T2, T3) β R β T1 is congruent to T2 and T2 is
congruent to T3 β T1 is congruent to T3 β (T1, T3) β R Therefore, R is an equivalence
relation Example 3 Let L be the set of all lines in a plane and R be the relation in L defined as
R = {(L1, L2) : L1 is perpendicular to L2} |
1 | 398-401 | Moreover, (T1, T2), (T2, T3) β R β T1 is congruent to T2 and T2 is
congruent to T3 β T1 is congruent to T3 β (T1, T3) β R Therefore, R is an equivalence
relation Example 3 Let L be the set of all lines in a plane and R be the relation in L defined as
R = {(L1, L2) : L1 is perpendicular to L2} Show that R is symmetric but neither
reflexive nor transitive |
1 | 399-402 | Therefore, R is an equivalence
relation Example 3 Let L be the set of all lines in a plane and R be the relation in L defined as
R = {(L1, L2) : L1 is perpendicular to L2} Show that R is symmetric but neither
reflexive nor transitive Solution R is not reflexive, as a line L1 can not be perpendicular to itself, i |
1 | 400-403 | Example 3 Let L be the set of all lines in a plane and R be the relation in L defined as
R = {(L1, L2) : L1 is perpendicular to L2} Show that R is symmetric but neither
reflexive nor transitive Solution R is not reflexive, as a line L1 can not be perpendicular to itself, i e |
1 | 401-404 | Show that R is symmetric but neither
reflexive nor transitive Solution R is not reflexive, as a line L1 can not be perpendicular to itself, i e , (L1, L1)
β R |
1 | 402-405 | Solution R is not reflexive, as a line L1 can not be perpendicular to itself, i e , (L1, L1)
β R R is symmetric as (L1, L2) β R
β
L1 is perpendicular to L2
β
L2 is perpendicular to L1
β
(L2, L1) β R |
1 | 403-406 | e , (L1, L1)
β R R is symmetric as (L1, L2) β R
β
L1 is perpendicular to L2
β
L2 is perpendicular to L1
β
(L2, L1) β R R is not transitive |
1 | 404-407 | , (L1, L1)
β R R is symmetric as (L1, L2) β R
β
L1 is perpendicular to L2
β
L2 is perpendicular to L1
β
(L2, L1) β R R is not transitive Indeed, if L1 is perpendicular to L2 and
L2 is perpendicular to L3, then L1 can never be perpendicular to
L3 |
1 | 405-408 | R is symmetric as (L1, L2) β R
β
L1 is perpendicular to L2
β
L2 is perpendicular to L1
β
(L2, L1) β R R is not transitive Indeed, if L1 is perpendicular to L2 and
L2 is perpendicular to L3, then L1 can never be perpendicular to
L3 In fact, L1 is parallel to L3, i |
1 | 406-409 | R is not transitive Indeed, if L1 is perpendicular to L2 and
L2 is perpendicular to L3, then L1 can never be perpendicular to
L3 In fact, L1 is parallel to L3, i e |
1 | 407-410 | Indeed, if L1 is perpendicular to L2 and
L2 is perpendicular to L3, then L1 can never be perpendicular to
L3 In fact, L1 is parallel to L3, i e , (L1, L2) β R, (L2, L3) β R but (L1, L3) β R |
1 | 408-411 | In fact, L1 is parallel to L3, i e , (L1, L2) β R, (L2, L3) β R but (L1, L3) β R Example 4 Show that the relation R in the set {1, 2, 3} given by R = {(1, 1), (2, 2),
(3, 3), (1, 2), (2, 3)} is reflexive but neither symmetric nor transitive |
1 | 409-412 | e , (L1, L2) β R, (L2, L3) β R but (L1, L3) β R Example 4 Show that the relation R in the set {1, 2, 3} given by R = {(1, 1), (2, 2),
(3, 3), (1, 2), (2, 3)} is reflexive but neither symmetric nor transitive Solution R is reflexive, since (1, 1), (2, 2) and (3, 3) lie in R |
1 | 410-413 | , (L1, L2) β R, (L2, L3) β R but (L1, L3) β R Example 4 Show that the relation R in the set {1, 2, 3} given by R = {(1, 1), (2, 2),
(3, 3), (1, 2), (2, 3)} is reflexive but neither symmetric nor transitive Solution R is reflexive, since (1, 1), (2, 2) and (3, 3) lie in R Also, R is not symmetric,
as (1, 2) β R but (2, 1) β R |
1 | 411-414 | Example 4 Show that the relation R in the set {1, 2, 3} given by R = {(1, 1), (2, 2),
(3, 3), (1, 2), (2, 3)} is reflexive but neither symmetric nor transitive Solution R is reflexive, since (1, 1), (2, 2) and (3, 3) lie in R Also, R is not symmetric,
as (1, 2) β R but (2, 1) β R Similarly, R is not transitive, as (1, 2) β R and (2, 3) β R
but (1, 3) β R |
1 | 412-415 | Solution R is reflexive, since (1, 1), (2, 2) and (3, 3) lie in R Also, R is not symmetric,
as (1, 2) β R but (2, 1) β R Similarly, R is not transitive, as (1, 2) β R and (2, 3) β R
but (1, 3) β R Example 5 Show that the relation R in the set Z of integers given by
R = {(a, b) : 2 divides a β b}
is an equivalence relation |
1 | 413-416 | Also, R is not symmetric,
as (1, 2) β R but (2, 1) β R Similarly, R is not transitive, as (1, 2) β R and (2, 3) β R
but (1, 3) β R Example 5 Show that the relation R in the set Z of integers given by
R = {(a, b) : 2 divides a β b}
is an equivalence relation Solution R is reflexive, as 2 divides (a β a) for all a β Z |
1 | 414-417 | Similarly, R is not transitive, as (1, 2) β R and (2, 3) β R
but (1, 3) β R Example 5 Show that the relation R in the set Z of integers given by
R = {(a, b) : 2 divides a β b}
is an equivalence relation Solution R is reflexive, as 2 divides (a β a) for all a β Z Further, if (a, b) β R, then
2 divides a β b |
1 | 415-418 | Example 5 Show that the relation R in the set Z of integers given by
R = {(a, b) : 2 divides a β b}
is an equivalence relation Solution R is reflexive, as 2 divides (a β a) for all a β Z Further, if (a, b) β R, then
2 divides a β b Therefore, 2 divides b β a |
1 | 416-419 | Solution R is reflexive, as 2 divides (a β a) for all a β Z Further, if (a, b) β R, then
2 divides a β b Therefore, 2 divides b β a Hence, (b, a) β R, which shows that R is
symmetric |
1 | 417-420 | Further, if (a, b) β R, then
2 divides a β b Therefore, 2 divides b β a Hence, (b, a) β R, which shows that R is
symmetric Similarly, if (a, b) β R and (b, c) β R, then a β b and b β c are divisible by
2 |
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