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Let Z be the set of integers and R be the relation defined in Z such that aRb if a – b is divisible by 3. Then R partitions the set Z into ______ pairwise disjoint subsets

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प्रश्न

Let Z be the set of integers and R be the relation defined in Z such that aRb if a – b is divisible by 3. Then R partitions the set Z into ______ pairwise disjoint subsets

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उत्तर

Let Z be the set of integers and R be the relation defined in Z such that aRb if a – b is divisible by 3. Then R partitions the set Z into three pairwise disjoint subsets

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अध्याय 1: Relations And Functions - Solved Examples [पृष्ठ १०]

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एनसीईआरटी एक्झांप्लर Mathematics Exemplar [English] Class 12
अध्याय 1 Relations And Functions
Solved Examples | Q 28 | पृष्ठ १०

संबंधित प्रश्न

Let A = {1, 2, 3,......, 9} and R be the relation in A × A defined by (a, b) R (c, d) if a + d = b + c for (a, b), (c, d) in A × A. Prove that R is an equivalence relation. Also, obtain the equivalence class [(2, 5)].


Let N denote the set of all natural numbers and R be the relation on N × N defined by (a, b) R (c, d) if ad (b + c) = bc (a + d). Show that R is an equivalence relation.


Determine whether the following relation is reflexive, symmetric and transitive:

Relation R in the set A = {1, 2, 3, ..., 13, 14} defined as R = {(x, y) : 3x – y = 0}.


Determine whether the following relation is reflexive, symmetric and transitive:

Relation R in the set N of natural numbers defined as R = {(x, y) : y = x + 5 and x < 4}.


Check whether the relation R in R defined by R = {(a, b) : a ≤ b3} is reflexive, symmetric or transitive.


Given an example of a relation. Which is symmetric but neither reflexive nor transitive.


Given an example of a relation. Which is reflexive and transitive but not symmetric.


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?


Given a non-empty set X, consider P(X), which is the set of all subsets of X. Define the relation R in P(X) as follows:

For subsets A, B in P(X), ARB if and only if A ⊂ B. Is R an equivalence relation on P(X)? Justify your answer.


Let A = {1, 2, 3}. Then number of relations containing (1, 2) and (1, 3) which are reflexive and symmetric but not transitive is ______.


The following relation is defined on the set of real numbers.

aRb if 1 + ab > 0

Find whether relation is reflexive, symmetric or transitive.


Let R be a relation on the set A of ordered pair of integers defined by (x, y) R (u, v) if xv = yu. Show that R is an equivalence relation.


Show that the relation R, defined in the set A of all polygons as R = {(P1, P2) : P1 and P2 have the same number of sides}, is an equivalence relation. What is the set of all elements in A related to the right-angled triangle T with sides 3, 4 and 5?


If R and S are transitive relations on a set A, then prove that R ∪ S may not be a transitive relation on A.


Write the domain of the relation R defined on the set Z of integers as follows:-
(a, b) ∈ R ⇔ a2 + b2 = 25


Define a symmetric relation ?


Define an equivalence relation ?


Let the relation R be defined on N by aRb iff 2a + 3b = 30. Then write R as a set of ordered pairs


Write the smallest equivalence relation on the set A = {1, 2, 3} ?


Mark the correct alternative in the following question:

The maximum number of equivalence relations on the set A = {1, 2, 3} is _______________ .


Show that the relation S in the set A = [x ∈ Z : 0 ≤ x ≤ 12] given by S = [(a, b) : a, b ∈ Z, ∣a − b∣ is divisible by 3] is an equivalence relation.


If A = {a, b, c}, B = (x , y} find A × A.


Let R be a relation on the set N of natural numbers defined by nRm if n divides m. Then R is ______.


For real numbers x and y, define xRy if and only if x – y + `sqrt(2)` is an irrational number. Then the relation R is ______.


Given A = {2, 3, 4}, B = {2, 5, 6, 7}. Construct an example of the following:
a mapping from B to A


Give an example of a map which is one-one but not onto


If a relation R on the set {1, 2, 3} be defined by R = {(1, 2)}, then R is ______.


Let R = {(3, 1), (1, 3), (3, 3)} be a relation defined on the set A = {1, 2, 3}. Then R is symmetric, transitive but not reflexive.


Every relation which is symmetric and transitive is also reflexive.


An integer m is said to be related to another integer n if m is a integral multiple of n. This relation in Z is reflexive, symmetric and transitive.


Sherlin and Danju are playing Ludo at home during Covid-19. While rolling the dice, Sherlin’s sister Raji observed and noted the possible outcomes of the throw every time belongs to set {1,2,3,4,5,6}. Let A be the set of players while B be the set of all possible outcomes.

A = {S, D}, B = {1,2,3,4,5,6}

  • Raji wants to know the number of relations possible from A to B. How many numbers of relations are possible?

An organization conducted a bike race under 2 different categories-boys and girls. Totally there were 250 participants. Among all of them finally, three from Category 1 and two from Category 2 were selected for the final race. Ravi forms two sets B and G with these participants for his college project. Let B = {b1,b2,b3} G={g1,g2} where B represents the set of boys selected and G the set of girls who were selected for the final race.

Ravi decides to explore these sets for various types of relations and functions.

  • Let R: B → B be defined by R = {(x, y): x and y are students of same sex}, Then this relation R is ____________.

Students of Grade 9, planned to plant saplings along straight lines, parallel to each other to one side of the playground ensuring that they had enough play area. Let us assume that they planted one of the rows of the saplings along the line y = x − 4. Let L be the set of all lines which are parallel on the ground and R be a relation on L.

Answer the following using the above information.

  • Let R = `{ ("L"_1, "L"_2) ∶ "L"_1 bot "L"_2  "where"  "L"_1, "L"_2 in "L" }` which of the following is true?

If A is a finite set consisting of n elements, then the number of reflexive relations on A is


Let A = {3, 5}. Then number of reflexive relations on A is ______.


If a relation R on the set {a, b, c} defined by R = {(b, b)}, then classify the relation.


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