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1 | 418-421 | 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 Now, a β c = (a β b) + (b β c) is even (Why |
1 | 419-422 | 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 Now, a β c = (a β b) + (b β c) is even (Why ) |
1 | 420-423 | Similarly, if (a, b) β R and (b, c) β R, then a β b and b β c are divisible by
2 Now, a β c = (a β b) + (b β c) is even (Why ) So, (a β c) is divisible by 2 |
1 | 421-424 | Now, a β c = (a β b) + (b β c) is even (Why ) So, (a β c) is divisible by 2 This
shows that R is transitive |
1 | 422-425 | ) So, (a β c) is divisible by 2 This
shows that R is transitive Thus, R is an equivalence relation in Z |
1 | 423-426 | So, (a β c) is divisible by 2 This
shows that R is transitive Thus, R is an equivalence relation in Z Fig 1 |
1 | 424-427 | This
shows that R is transitive Thus, R is an equivalence relation in Z Fig 1 1
Rationalised 2023-24
MATHEMATICS
4
In Example 5, note that all even integers are related to zero, as (0, Β± 2), (0, Β± 4)
etc |
1 | 425-428 | Thus, R is an equivalence relation in Z Fig 1 1
Rationalised 2023-24
MATHEMATICS
4
In Example 5, note that all even integers are related to zero, as (0, Β± 2), (0, Β± 4)
etc , lie in R and no odd integer is related to 0, as (0, Β± 1), (0, Β± 3) etc |
1 | 426-429 | Fig 1 1
Rationalised 2023-24
MATHEMATICS
4
In Example 5, note that all even integers are related to zero, as (0, Β± 2), (0, Β± 4)
etc , lie in R and no odd integer is related to 0, as (0, Β± 1), (0, Β± 3) etc , do not lie in R |
1 | 427-430 | 1
Rationalised 2023-24
MATHEMATICS
4
In Example 5, note that all even integers are related to zero, as (0, Β± 2), (0, Β± 4)
etc , lie in R and no odd integer is related to 0, as (0, Β± 1), (0, Β± 3) etc , do not lie in R Similarly, all odd integers are related to one and no even integer is related to one |
1 | 428-431 | , lie in R and no odd integer is related to 0, as (0, Β± 1), (0, Β± 3) etc , do not lie in R Similarly, all odd integers are related to one and no even integer is related to one Therefore, the set E of all even integers and the set O of all odd integers are subsets of
Z satisfying following conditions:
(i)
All elements of E are related to each other and all elements of O are related to
each other |
1 | 429-432 | , do not lie in R Similarly, all odd integers are related to one and no even integer is related to one Therefore, the set E of all even integers and the set O of all odd integers are subsets of
Z satisfying following conditions:
(i)
All elements of E are related to each other and all elements of O are related to
each other (ii)
No element of E is related to any element of O and vice-versa |
1 | 430-433 | Similarly, all odd integers are related to one and no even integer is related to one Therefore, the set E of all even integers and the set O of all odd integers are subsets of
Z satisfying following conditions:
(i)
All elements of E are related to each other and all elements of O are related to
each other (ii)
No element of E is related to any element of O and vice-versa (iii)
E and O are disjoint and Z = E βͺ O |
1 | 431-434 | Therefore, the set E of all even integers and the set O of all odd integers are subsets of
Z satisfying following conditions:
(i)
All elements of E are related to each other and all elements of O are related to
each other (ii)
No element of E is related to any element of O and vice-versa (iii)
E and O are disjoint and Z = E βͺ O The subset E is called the equivalence class containing zero and is denoted by
[0] |
1 | 432-435 | (ii)
No element of E is related to any element of O and vice-versa (iii)
E and O are disjoint and Z = E βͺ O The subset E is called the equivalence class containing zero and is denoted by
[0] Similarly, O is the equivalence class containing 1 and is denoted by [1] |
1 | 433-436 | (iii)
E and O are disjoint and Z = E βͺ O The subset E is called the equivalence class containing zero and is denoted by
[0] Similarly, O is the equivalence class containing 1 and is denoted by [1] Note that
[0] β [1], [0] = [2r] and [1] = [2r + 1], r β Z |
1 | 434-437 | The subset E is called the equivalence class containing zero and is denoted by
[0] Similarly, O is the equivalence class containing 1 and is denoted by [1] Note that
[0] β [1], [0] = [2r] and [1] = [2r + 1], r β Z Infact, what we have seen above is true
for an arbitrary equivalence relation R in a set X |
1 | 435-438 | Similarly, O is the equivalence class containing 1 and is denoted by [1] Note that
[0] β [1], [0] = [2r] and [1] = [2r + 1], r β Z Infact, what we have seen above is true
for an arbitrary equivalence relation R in a set X Given an arbitrary equivalence
relation R in an arbitrary set X, R divides X into mutually disjoint subsets Ai called
partitions or subdivisions of X satisfying:
(i)
all elements of Ai are related to each other, for all i |
1 | 436-439 | Note that
[0] β [1], [0] = [2r] and [1] = [2r + 1], r β Z Infact, what we have seen above is true
for an arbitrary equivalence relation R in a set X Given an arbitrary equivalence
relation R in an arbitrary set X, R divides X into mutually disjoint subsets Ai called
partitions or subdivisions of X satisfying:
(i)
all elements of Ai are related to each other, for all i (ii)
no element of Ai is related to any element of Aj , i β j |
1 | 437-440 | Infact, what we have seen above is true
for an arbitrary equivalence relation R in a set X Given an arbitrary equivalence
relation R in an arbitrary set X, R divides X into mutually disjoint subsets Ai called
partitions or subdivisions of X satisfying:
(i)
all elements of Ai are related to each other, for all i (ii)
no element of Ai is related to any element of Aj , i β j (iii)
βͺ Aj = X and Ai β© Aj = Ο, i β j |
1 | 438-441 | Given an arbitrary equivalence
relation R in an arbitrary set X, R divides X into mutually disjoint subsets Ai called
partitions or subdivisions of X satisfying:
(i)
all elements of Ai are related to each other, for all i (ii)
no element of Ai is related to any element of Aj , i β j (iii)
βͺ Aj = X and Ai β© Aj = Ο, i β j The subsets Ai are called equivalence classes |
1 | 439-442 | (ii)
no element of Ai is related to any element of Aj , i β j (iii)
βͺ Aj = X and Ai β© Aj = Ο, i β j The subsets Ai are called equivalence classes The interesting part of the situation
is that we can go reverse also |
1 | 440-443 | (iii)
βͺ Aj = X and Ai β© Aj = Ο, i β j The subsets Ai are called equivalence classes The interesting part of the situation
is that we can go reverse also For example, consider a subdivision of the set Z given
by three mutually disjoint subsets A1, A2 and A3 whose union is Z with
A1 = {x β Z : x is a multiple of 3} = { |
1 | 441-444 | The subsets Ai are called equivalence classes The interesting part of the situation
is that we can go reverse also For example, consider a subdivision of the set Z given
by three mutually disjoint subsets A1, A2 and A3 whose union is Z with
A1 = {x β Z : x is a multiple of 3} = { , β 6, β 3, 0, 3, 6, |
1 | 442-445 | The interesting part of the situation
is that we can go reverse also For example, consider a subdivision of the set Z given
by three mutually disjoint subsets A1, A2 and A3 whose union is Z with
A1 = {x β Z : x is a multiple of 3} = { , β 6, β 3, 0, 3, 6, }
A2 = {x β Z : x β 1 is a multiple of 3} = { |
1 | 443-446 | For example, consider a subdivision of the set Z given
by three mutually disjoint subsets A1, A2 and A3 whose union is Z with
A1 = {x β Z : x is a multiple of 3} = { , β 6, β 3, 0, 3, 6, }
A2 = {x β Z : x β 1 is a multiple of 3} = { , β 5, β 2, 1, 4, 7, |
1 | 444-447 | , β 6, β 3, 0, 3, 6, }
A2 = {x β Z : x β 1 is a multiple of 3} = { , β 5, β 2, 1, 4, 7, }
A3 = {x β Z : x β 2 is a multiple of 3} = { |
1 | 445-448 | }
A2 = {x β Z : x β 1 is a multiple of 3} = { , β 5, β 2, 1, 4, 7, }
A3 = {x β Z : x β 2 is a multiple of 3} = { , β 4, β 1, 2, 5, 8, |
1 | 446-449 | , β 5, β 2, 1, 4, 7, }
A3 = {x β Z : x β 2 is a multiple of 3} = { , β 4, β 1, 2, 5, 8, }
Define a relation R in Z given by R = {(a, b) : 3 divides a β b} |
1 | 447-450 | }
A3 = {x β Z : x β 2 is a multiple of 3} = { , β 4, β 1, 2, 5, 8, }
Define a relation R in Z given by R = {(a, b) : 3 divides a β b} Following the
arguments similar to those used in Example 5, we can show that R is an equivalence
relation |
1 | 448-451 | , β 4, β 1, 2, 5, 8, }
Define a relation R in Z given by R = {(a, b) : 3 divides a β b} Following the
arguments similar to those used in Example 5, we can show that R is an equivalence
relation Also, A1 coincides with the set of all integers in Z which are related to zero, A2
coincides with the set of all integers which are related to 1 and A3 coincides with the
set of all integers in Z which are related to 2 |
1 | 449-452 | }
Define a relation R in Z given by R = {(a, b) : 3 divides a β b} Following the
arguments similar to those used in Example 5, we can show that R is an equivalence
relation Also, A1 coincides with the set of all integers in Z which are related to zero, A2
coincides with the set of all integers which are related to 1 and A3 coincides with the
set of all integers in Z which are related to 2 Thus, A1 = [0], A2 = [1] and A3 = [2] |
1 | 450-453 | Following the
arguments similar to those used in Example 5, we can show that R is an equivalence
relation Also, A1 coincides with the set of all integers in Z which are related to zero, A2
coincides with the set of all integers which are related to 1 and A3 coincides with the
set of all integers in Z which are related to 2 Thus, A1 = [0], A2 = [1] and A3 = [2] In fact, A1 = [3r], A2 = [3r + 1] and A3 = [3r + 2], for all r β Z |
1 | 451-454 | Also, A1 coincides with the set of all integers in Z which are related to zero, A2
coincides with the set of all integers which are related to 1 and A3 coincides with the
set of all integers in Z which are related to 2 Thus, A1 = [0], A2 = [1] and A3 = [2] In fact, A1 = [3r], A2 = [3r + 1] and A3 = [3r + 2], for all r β Z Example 6 Let R be the relation defined in the set A = {1, 2, 3, 4, 5, 6, 7} by
R = {(a, b) : both a and b are either odd or even} |
1 | 452-455 | Thus, A1 = [0], A2 = [1] and A3 = [2] In fact, A1 = [3r], A2 = [3r + 1] and A3 = [3r + 2], for all r β Z Example 6 Let R be the relation defined in the set A = {1, 2, 3, 4, 5, 6, 7} by
R = {(a, b) : both a and b are either odd or even} Show that R is an equivalence
relation |
1 | 453-456 | In fact, A1 = [3r], A2 = [3r + 1] and A3 = [3r + 2], for all r β Z Example 6 Let R be the relation defined in the set A = {1, 2, 3, 4, 5, 6, 7} by
R = {(a, b) : both a and b are either odd or even} Show that R is an equivalence
relation Further, show that all the elements of the subset {1, 3, 5, 7} are related to each
other and all the elements of the subset {2, 4, 6} are related to each other, but no
element of the subset {1, 3, 5, 7} is related to any element of the subset {2, 4, 6} |
1 | 454-457 | Example 6 Let R be the relation defined in the set A = {1, 2, 3, 4, 5, 6, 7} by
R = {(a, b) : both a and b are either odd or even} Show that R is an equivalence
relation Further, show that all the elements of the subset {1, 3, 5, 7} are related to each
other and all the elements of the subset {2, 4, 6} are related to each other, but no
element of the subset {1, 3, 5, 7} is related to any element of the subset {2, 4, 6} Rationalised 2023-24
RELATIONS AND FUNCTIONS
5
Solution Given any element a in A, both a and a must be either odd or even, so
that (a, a) β R |
1 | 455-458 | Show that R is an equivalence
relation Further, show that all the elements of the subset {1, 3, 5, 7} are related to each
other and all the elements of the subset {2, 4, 6} are related to each other, but no
element of the subset {1, 3, 5, 7} is related to any element of the subset {2, 4, 6} Rationalised 2023-24
RELATIONS AND FUNCTIONS
5
Solution Given any element a in A, both a and a must be either odd or even, so
that (a, a) β R Further, (a, b) β R β both a and b must be either odd or even
β (b, a) β R |
1 | 456-459 | Further, show that all the elements of the subset {1, 3, 5, 7} are related to each
other and all the elements of the subset {2, 4, 6} are related to each other, but no
element of the subset {1, 3, 5, 7} is related to any element of the subset {2, 4, 6} Rationalised 2023-24
RELATIONS AND FUNCTIONS
5
Solution Given any element a in A, both a and a must be either odd or even, so
that (a, a) β R Further, (a, b) β R β both a and b must be either odd or even
β (b, a) β R Similarly, (a, b) β R and (b, c) β R β all elements a, b, c, must be
either even or odd simultaneously β (a, c) β R |
1 | 457-460 | Rationalised 2023-24
RELATIONS AND FUNCTIONS
5
Solution Given any element a in A, both a and a must be either odd or even, so
that (a, a) β R Further, (a, b) β R β both a and b must be either odd or even
β (b, a) β R Similarly, (a, b) β R and (b, c) β R β all elements a, b, c, must be
either even or odd simultaneously β (a, c) β R Hence, R is an equivalence relation |
1 | 458-461 | Further, (a, b) β R β both a and b must be either odd or even
β (b, a) β R Similarly, (a, b) β R and (b, c) β R β all elements a, b, c, must be
either even or odd simultaneously β (a, c) β R Hence, R is an equivalence relation Further, all the elements of {1, 3, 5, 7} are related to each other, as all the elements
of this subset are odd |
1 | 459-462 | Similarly, (a, b) β R and (b, c) β R β all elements a, b, c, must be
either even or odd simultaneously β (a, c) β R Hence, R is an equivalence relation Further, all the elements of {1, 3, 5, 7} are related to each other, as all the elements
of this subset are odd Similarly, all the elements of the subset {2, 4, 6} are related to
each other, as all of them are even |
1 | 460-463 | Hence, R is an equivalence relation Further, all the elements of {1, 3, 5, 7} are related to each other, as all the elements
of this subset are odd Similarly, all the elements of the subset {2, 4, 6} are related to
each other, as all of them are even Also, no element of the subset {1, 3, 5, 7} can be
related to any element of {2, 4, 6}, as elements of {1, 3, 5, 7} are odd, while elements
of {2, 4, 6} are even |
1 | 461-464 | Further, all the elements of {1, 3, 5, 7} are related to each other, as all the elements
of this subset are odd Similarly, all the elements of the subset {2, 4, 6} are related to
each other, as all of them are even Also, no element of the subset {1, 3, 5, 7} can be
related to any element of {2, 4, 6}, as elements of {1, 3, 5, 7} are odd, while elements
of {2, 4, 6} are even EXERCISE 1 |
1 | 462-465 | Similarly, all the elements of the subset {2, 4, 6} are related to
each other, as all of them are even Also, no element of the subset {1, 3, 5, 7} can be
related to any element of {2, 4, 6}, as elements of {1, 3, 5, 7} are odd, while elements
of {2, 4, 6} are even EXERCISE 1 1
1 |
1 | 463-466 | Also, no element of the subset {1, 3, 5, 7} can be
related to any element of {2, 4, 6}, as elements of {1, 3, 5, 7} are odd, while elements
of {2, 4, 6} are even EXERCISE 1 1
1 Determine whether each of the following relations are reflexive, symmetric and
transitive:
(i) Relation R in the set A = {1, 2, 3, |
1 | 464-467 | EXERCISE 1 1
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) : 3x β 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 β y is an 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}
2 |
1 | 465-468 | 1
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) : 3x β 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 β y is an 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}
2 Show that the relation R in the set R of real numbers, defined as
R = {(a, b) : a β€ b2} is neither reflexive nor symmetric nor transitive |
1 | 466-469 | 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) : 3x β 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 β y is an 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}
2 Show that the relation R in the set R of real numbers, defined as
R = {(a, b) : a β€ b2} is neither reflexive nor symmetric nor transitive 3 |
1 | 467-470 | , 13, 14} defined as
R = {(x, y) : 3x β 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 β y is an 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}
2 Show that the relation R in the set R of real numbers, defined as
R = {(a, b) : a β€ b2} is neither reflexive nor symmetric nor transitive 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 |
1 | 468-471 | Show that the relation R in the set R of real numbers, defined as
R = {(a, b) : a β€ b2} is neither reflexive nor symmetric nor transitive 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 4 |
1 | 469-472 | 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 4 Show that the relation R in R defined as R = {(a, b) : a β€ b}, is reflexive and
transitive but not symmetric |
1 | 470-473 | 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 4 Show that the relation R in R defined as R = {(a, b) : a β€ b}, is reflexive and
transitive but not symmetric 5 |
1 | 471-474 | 4 Show that the relation R in R defined as R = {(a, b) : a β€ b}, is reflexive and
transitive but not symmetric 5 Check whether the relation R in R defined by R = {(a, b) : a β€ b3} is reflexive,
symmetric or transitive |
1 | 472-475 | Show that the relation R in R defined as R = {(a, b) : a β€ b}, is reflexive and
transitive but not symmetric 5 Check whether the relation R in R defined by R = {(a, b) : a β€ b3} is reflexive,
symmetric or transitive Rationalised 2023-24
MATHEMATICS
6
6 |
1 | 473-476 | 5 Check whether the relation R in R defined by R = {(a, b) : a β€ b3} is reflexive,
symmetric or transitive Rationalised 2023-24
MATHEMATICS
6
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 |
1 | 474-477 | Check whether the relation R in R defined by R = {(a, b) : a β€ b3} is reflexive,
symmetric or transitive Rationalised 2023-24
MATHEMATICS
6
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 7 |
1 | 475-478 | Rationalised 2023-24
MATHEMATICS
6
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 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 |
1 | 476-479 | 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 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 8 |
1 | 477-480 | 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 8 Show that the relation R in the set A = {1, 2, 3, 4, 5} given by
R = {(a, b) : |a β b| is even}, is an equivalence relation |
1 | 478-481 | 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 8 Show that the relation R in the set A = {1, 2, 3, 4, 5} given by
R = {(a, b) : |a β b| is even}, 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 |
1 | 479-482 | 8 Show that the relation R in the set A = {1, 2, 3, 4, 5} given by
R = {(a, b) : |a β b| is even}, 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} |
1 | 480-483 | Show that the relation R in the set A = {1, 2, 3, 4, 5} given by
R = {(a, b) : |a β b| is even}, 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} 9 |
1 | 481-484 | 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} 9 Show that each of the relation R in the set A = {x β Z : 0 β€ x β€ 12}, given by
(i) R = {(a, b) : |a β b| is a multiple of 4}
(ii) R = {(a, b) : a = b}
is an equivalence relation |
1 | 482-485 | But no element of {1, 3, 5} is related to any element of {2, 4} 9 Show that each of the relation R in the set A = {x β Z : 0 β€ x β€ 12}, given by
(i) R = {(a, b) : |a β b| is a multiple of 4}
(ii) R = {(a, b) : a = b}
is an equivalence relation Find the set of all elements related to 1 in each case |
1 | 483-486 | 9 Show that each of the relation R in the set A = {x β Z : 0 β€ x β€ 12}, given by
(i) R = {(a, b) : |a β b| is a multiple of 4}
(ii) R = {(a, b) : a = b}
is an equivalence relation Find the set of all elements related to 1 in each case 10 |
1 | 484-487 | Show that each of the relation R in the set A = {x β Z : 0 β€ x β€ 12}, given by
(i) R = {(a, b) : |a β b| is a multiple of 4}
(ii) R = {(a, b) : a = b}
is an equivalence relation Find the set of all elements related to 1 in each case 10 Give an example of a relation |
1 | 485-488 | Find the set of all elements related to 1 in each case 10 Give an example of a relation Which is
(i) Symmetric but neither reflexive nor transitive |
1 | 486-489 | 10 Give an example of a relation Which is
(i) Symmetric but neither reflexive nor transitive (ii) Transitive but neither reflexive nor symmetric |
1 | 487-490 | Give 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 |
1 | 488-491 | 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 |
1 | 489-492 | (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 |
1 | 490-493 | (iii) Reflexive and symmetric but not transitive (iv) Reflexive and transitive but not symmetric (v) Symmetric and transitive but not reflexive 11 |
1 | 491-494 | (iv) Reflexive and transitive but not symmetric (v) Symmetric and transitive but not reflexive 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 |
1 | 492-495 | (v) Symmetric and transitive but not reflexive 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 P β (0, 0) is the circle passing through P with origin as
centre |
1 | 493-496 | 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 P β (0, 0) is the circle passing through P with origin as
centre 12 |
1 | 494-497 | 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 P β (0, 0) is the circle passing through P with origin as
centre 12 Show that the relation R defined in the set A of all triangles as R = {(T1, T2) : T1
is similar to T2}, is equivalence relation |
1 | 495-498 | Further, show that the set of
all points related to a point P β (0, 0) is the circle passing through P with origin as
centre 12 Show that the relation R defined in the set A of all triangles as R = {(T1, T2) : T1
is similar to T2}, is equivalence relation Consider three right angle triangles T1
with sides 3, 4, 5, T2 with sides 5, 12, 13 and T3 with sides 6, 8, 10 |
1 | 496-499 | 12 Show that the relation R defined in the set A of all triangles as R = {(T1, T2) : T1
is similar to T2}, is equivalence relation Consider three right angle triangles T1
with sides 3, 4, 5, T2 with sides 5, 12, 13 and T3 with sides 6, 8, 10 Which
triangles among T1, T2 and T3 are related |
1 | 497-500 | Show that the relation R defined in the set A of all triangles as R = {(T1, T2) : T1
is similar to T2}, is equivalence relation Consider three right angle triangles T1
with sides 3, 4, 5, T2 with sides 5, 12, 13 and T3 with sides 6, 8, 10 Which
triangles among T1, T2 and T3 are related 13 |
1 | 498-501 | Consider three right angle triangles T1
with sides 3, 4, 5, T2 with sides 5, 12, 13 and T3 with sides 6, 8, 10 Which
triangles among T1, T2 and T3 are related 13 Show that the relation R defined in the set A of all polygons as R = {(P1, P2) :
P1 and P2 have same number of sides}, is an equivalence relation |
1 | 499-502 | Which
triangles among T1, T2 and T3 are related 13 Show that the relation R defined in the set A of all polygons as R = {(P1, P2) :
P1 and P2 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 |
1 | 500-503 | 13 Show that the relation R defined in the set A of all polygons as R = {(P1, P2) :
P1 and P2 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 14 |
1 | 501-504 | Show that the relation R defined in the set A of all polygons as R = {(P1, P2) :
P1 and P2 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 14 Let L be the set of all lines in XY plane and R be the relation in L defined as
R = {(L1, L2) : L1 is parallel to L2} |
1 | 502-505 | What is the
set of all elements in A related to the right angle triangle T with sides 3, 4 and 5 14 Let L be the set of all lines in XY plane and R be the relation in L defined as
R = {(L1, L2) : L1 is parallel to L2} Show that R is an equivalence relation |
1 | 503-506 | 14 Let L be the set of all lines in XY plane and R be the relation in L defined as
R = {(L1, L2) : L1 is parallel to L2} Show that R is an equivalence relation Find
the set of all lines related to the line y = 2x + 4 |
1 | 504-507 | Let L be the set of all lines in XY plane and R be the relation in L defined as
R = {(L1, L2) : L1 is parallel to L2} Show that R is an equivalence relation Find
the set of all lines related to the line y = 2x + 4 Rationalised 2023-24
RELATIONS AND FUNCTIONS
7
15 |
1 | 505-508 | Show that R is an equivalence relation Find
the set of all lines related to the line y = 2x + 4 Rationalised 2023-24
RELATIONS AND FUNCTIONS
7
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)} |
1 | 506-509 | Find
the set of all lines related to the line y = 2x + 4 Rationalised 2023-24
RELATIONS AND FUNCTIONS
7
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 |
1 | 507-510 | Rationalised 2023-24
RELATIONS AND FUNCTIONS
7
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 |
1 | 508-511 | 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 |
1 | 509-512 | 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 |
1 | 510-513 | (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 |
1 | 511-514 | (B) R is reflexive and transitive but not symmetric (C) R is symmetric and transitive but not reflexive (D) R is an equivalence relation 16 |
1 | 512-515 | (C) R is symmetric and transitive but not reflexive (D) R is an equivalence relation 16 Let R be the relation in the set N given by R = {(a, b) : a = b β 2, b > 6} |
1 | 513-516 | (D) R is an equivalence relation 16 Let R be the relation in the set N given by R = {(a, b) : a = b β 2, b > 6} Choose
the correct answer |
1 | 514-517 | 16 Let R be the relation in the set N given by R = {(a, b) : a = b β 2, b > 6} Choose
the correct answer (A) (2, 4) β R
(B) (3, 8) β R
(C) (6, 8) β R
(D) (8, 7) β R
1 |
1 | 515-518 | Let R be the relation in the set N given by R = {(a, b) : a = b β 2, b > 6} Choose
the correct answer (A) (2, 4) β R
(B) (3, 8) β R
(C) (6, 8) β R
(D) (8, 7) β R
1 3 Types of Functions
The notion of a function along with some special functions like identity function, constant
function, polynomial function, rational function, modulus function, signum function etc |
1 | 516-519 | Choose
the correct answer (A) (2, 4) β R
(B) (3, 8) β R
(C) (6, 8) β R
(D) (8, 7) β R
1 3 Types of Functions
The notion of a function along with some special functions like identity function, constant
function, polynomial function, rational function, modulus function, signum function etc along with their graphs have been given in Class XI |
1 | 517-520 | (A) (2, 4) β R
(B) (3, 8) β R
(C) (6, 8) β R
(D) (8, 7) β R
1 3 Types of Functions
The notion of a function along with some special functions like identity function, constant
function, polynomial function, rational function, modulus function, signum function etc along with their graphs have been given in Class XI Addition, subtraction, multiplication and division of two functions have also been
studied |
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