Advertisements
Advertisements
प्रश्न
Give an example of a relation which is reflexive and transitive but not symmetric ?
उत्तर
Let a relation R be defined on a set R.
R = {(a, b) : a3 ≥ b3}
Therefore, (a, a) ∈ R. ......[because a3 = a3]
∴ R is Reflexive ..... [because 23 ≥ 13]
Here, (2, 1) ∈ R .....[because 13 ≥ 23]
∴ R is not symmetric.
Now, let (a, b) and (b, c) ∈ R.
∴ R is transitive.
Hence, the relation R is self-equivalent and transitive but not symmetric.
APPEARS IN
संबंधित प्रश्न
Show that the relation R in R defined as R = {(a, b): a ≤ b}, is reflexive and transitive but not symmetric.
Show that each of the relation R in the set A= {x ∈ Z : 0 ≤ x ≤ = 12} given by R = {(a, b) : |a - b| is a multiple of 4} is an equivalence relation. Find the set of all elements related to 1 in each case.
Show that each of the relation R in the set A= {x ∈ Z : 0 ≤ x ≤ = 12} given by R = {(a,b) : a = b} is an equivalence relation. Find the set of all elements related to 1 in each case.
Given an example of a relation. Which is Reflexive and transitive but not symmetric.
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.
The binary operation *: R x R → R is defined as a *b = 2a + b Find (2 * 3)*4
Test whether the following relation R2 is (i) reflexive (ii) symmetric and (iii) transitive:
R2 on Z defined by (a, b) ∈ R2 ⇔ |a – b| ≤ 5
Let A = {1, 2, 3}, and let R1 = {(1, 1), (1, 3), (3, 1), (2, 2), (2, 1), (3, 3)}, R2 = {(2, 2), (3, 1), (1, 3)}, R3 = {(1, 3), (3, 3)}. Find whether or not each of the relations R1, R2, R3 on A is (i) reflexive (ii) symmetric (iii) transitive.
Defines a relation on N:
x + 4y = 10, x, y ∈ N
Determine the above relation is reflexive, symmetric and transitive.
Prove that the relation R on Z defined by
(a, b) ∈ R ⇔ a − b is divisible by 5
is an equivalence relation on Z.
If R is a symmetric relation on a set A, then write a relation between R and R−1.
If A = {2, 3, 4}, B = {1, 3, 7} and R = {(x, y) : x ∈ A, y ∈ B and x < y} is a relation from A to B, then write R−1.
Define a symmetric relation ?
Define an equivalence relation ?
Let R = {(a, a3) : a is a prime number less than 5} be a relation. Find the range of R.
R is a relation from {11, 12, 13} to {8, 10, 12} defined by y = x − 3. Then, R−1 is ______________ .
If A = {1, 2, 3}, then a relation R = {(2, 3)} on A is _____________ .
Mark the correct alternative in the following question:
Let R be a relation on the set N of natural numbers defined by nRm if n divides m. Then, R is _____________ .
Mark the correct alternative in the following question:
Consider a non-empty set consisting of children in a family and a relation R defined as aRb if a is brother of b. Then, R is _____________ .
Show that the relation R on the set Z of integers, given by R = {(a,b):2divides (a - b)} is an equivalence relation.
If A = {a, b, c}, B = (x , y} find B × A.
Let A = {1, 2, 3, 4}, B = {4, 5, 6}, C = {5, 6}. Find A × (B ∪ C).
Write the relation in the Roster form and hence find its domain and range :
R1 = {(a, a2) / a is prime number less than 15}
R = {(a, b) / b = a + 1, a ∈ Z, 0 < a < 5}. Find the Range of R.
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 neither one-one nor onto
Let A = { 2, 3, 6 } Which of the following relations on A are reflexive?
Let A = {1, 2, 3, 4, 5, 6} Which of the following partitions of A correspond to an equivalence relation on A?
A relation R on a non – empty set A is an equivalence relation if it is ____________.
Let T be the set of all triangles in the Euclidean plane, and let a relation R on T be defined as aRb if a is congruent to b ∀ a, b ∈ T. Then R is ____________.
Let A = {1, 2, 3, …. n} and B = {a, b}. Then the number of surjections from A into B is ____________.
The number of surjective functions from A to B where A = {1, 2, 3, 4} and B = {a, b} is
A relation in a set 'A' is known as empty relation:-
If f(x + 2a) = f(x – 2a), then f(x) is:
Let a set A = A1 ∪ A2 ∪ ... ∪ Ak, where Ai ∩ Aj = Φ for i ≠ j, 1 ≤ i, j ≤ k. Define the relation R from A to A by R = {(x, y): y ∈ Ai if and only if x ∈ Ai, 1 ≤ i ≤ k}. Then, R is ______.
Let R = {(x, y) : x, y ∈ N and x2 – 4xy + 3y2 = 0}, where N is the set of all natural numbers. Then the relation R is ______.
Read the following passage:
|
An organization conducted bike race under two different categories – Boys and Girls. There were 28 participants in all. 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. |
Based on the above information, answer the following questions:
- How many relations are possible from B to G? (1)
- Among all the possible relations from B to G, how many functions can be formed from B to G? (1)
- Let R : B `rightarrow` B be defined by R = {(x, y) : x and y are students of the same sex}. Check if R is an equivalence relation. (2)
OR
A function f : B `rightarrow` G be defined by f = {(b1, g1), (b2, g2), (b3, g1)}. Check if f is bijective. Justify your answer. (2)
