Advertisements
Advertisements
प्रश्न
Let A be the set of all human beings in a town at a particular time. Determine whether the following relation is reflexive, symmetric and transitive:
R = {(x, y) : x and y work at the same place}
Advertisements
उत्तर
(i) Reflexivity:
Let x be an arbitrary element of R.
Then, x ∈ R
⇒ x and x work at the same place,
Which is true since they are the same.
⇒ (x, x) ∈ R
∴ R is a reflexive relation.
(ii) Symmetry:
Let (x, y) ∈ R
⇒ x and y work at the same place.
⇒ y and x work at the same place.
⇒ (y, x) ∈ R
∴ R is a symmetric relation.
(iii) Transitivity:
Let (x, y) ∈ R and (y, z) ∈ R.
Then, x and y work at the same place.
y and z also work at the same place.
⇒ x, y and z all work at the same place.
⇒ x and z work at the same place.
⇒ (x, z) ∈ R
∴ R is a transitive relation.
Hence, R is reflexive, symmetric and transitive.
APPEARS IN
संबंधित प्रश्न
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 Z of all integers defined as R = {(x, y) : x – y is an integer}.
Show that the relation R in R defined as R = {(a, b) : a ≤ b}, is reflexive and transitive but not symmetric.
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.
Given an example of a relation. Which is reflexive and symmetric but not transitive.
Given an example of a relation. Which is reflexive and transitive but not symmetric.
Given an example of a relation. Which is symmetric and transitive but not reflexive.
Show that the relation R defined in the set A of all triangles as R = {(T1, T2) : T1 is similar to T2}, is an 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, and 10. Which triangles among T1, T2 and T3 are related?
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.
Let A = {1, 2, 3}. Then number of equivalence relations containing (1, 2) is ______.
Let A = {x ∈ Z : 0 ≤ x ≤ 12}. Show that R = {(a, b) : a, b ∈ A, |a – b| is divisible by 4}is an equivalence relation. Find the set of all elements related to 1. Also write the equivalence class [2]
The following relation is defined on the set of real numbers.
aRb if a – b > 0
Find whether relation is reflexive, symmetric or transitive.
The following relation is defined on the set of real numbers. aRb if |a| ≤ b
Find whether relation is reflexive, symmetric or transitive.
If A = {1, 2, 3, 4} define relations on A which have properties of being reflexive, symmetric and transitive ?
An integer m is said to be related to another integer n if m is a multiple of n. Check if the relation is symmetric, reflexive and transitive.
Give an example of a relation which is reflexive and transitive but not symmetric?
Defines a relation on N:
xy is square of an integer, 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.
Show that the relation R on 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.
If R and S are transitive relations on a set A, then prove that R ∪ S may not be a transitive relation on A.
Let R = {(x, y) : |x2 − y2| <1) be a relation on set A = {1, 2, 3, 4, 5}. Write R as a set of ordered pairs.
If A = {3, 5, 7} and B = {2, 4, 9} and R is a relation given by "is less than", write R as a set ordered pairs.
Let R be a relation on the set N given by
R = {(a, b) : a = b − 2, b > 6}. Then,
The relation 'R' in N × N such that
(a, b) R (c, d) ⇔ a + d = b + c is ______________ .
Let R be a relation on N defined by x + 2y = 8. The domain of R is _______________ .
Mark the correct alternative in the following question:
For real numbers x and y, define xRy if `x-y+sqrt2` is an irrational number. Then the relation R is ___________ .
Write the relation in the Roster form and hence find its domain and range:
R2 = `{("a", 1/"a") "/" 0 < "a" ≤ 5, "a" ∈ "N"}`
Let n be a fixed positive integer. Define a relation R in Z as follows: ∀ a, b ∈ Z, aRb if and only if a – b is divisible by n. Show that R is an equivalance relation
If A = {1, 2, 3, 4 }, define relations on A which have properties of being:
reflexive, transitive but not symmetric
The following defines a relation on N:
x + 4y = 10 x, y ∈ N.
Determine which of the above relations are reflexive, symmetric and transitive.
The maximum number of equivalence relations on the set A = {1, 2, 3} are ______.
Which of the following is not an equivalence relation on I, the set of integers: x, y
Let `"f"("x") = ("x" - 1)/("x" + 1),` then f(f(x)) is ____________.
Total number of equivalence relations defined in the set S = {a, b, c} is ____________.
Find: `int (x + 1)/((x^2 + 1)x) dx`
In a group of 52 persons, 16 drink tea but not coffee, while 33 drink tea. How many persons drink coffee but not tea?
Statement 1: The intersection of two equivalence relations is always an equivalence relation.
Statement 2: The Union of two equivalence relations is always an equivalence relation.
Which one of the following is correct?
