Advertisements
Advertisements
प्रश्न
Test whether the following relation R3 is (i) reflexive (ii) symmetric and (iii) transitive:
R3 on R is defined by (a, b) ∈ R3 `⇔` a2 – 4ab + 3b2 = 0.
Advertisements
उत्तर
Reflexivity:-
Let a be an arbitrary element of R3.
Then, a ∈ R3
⇒ a2 − 4a × a + 3a2 = 0
So, R3 is reflexive.
Symmetry:-
Let (a, b) ∈ R3
⇒ a2 − 4ab + 3b2 = 0
But b2 − 4ba + 3a2 ≠ 0 for all a, b ∈ R
So, R3 is not symmetric.
Transitivity:-
Let (1, 2) ∈ R3 and (2, 3) ∈ R3
⇒ 1 − 8 + 6 = 0 and 4 − 24 + 27 = 0
−7 + 6 = 0 and −20 + 27 = 0
−1 ≠ 0 and 7 ≠ 0
So, R3 is not transitive.
APPEARS IN
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.
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.
Show that 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.
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?
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 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}
Defines a relation on N :
x > y, x, y ∈ N
Determine the above relation is reflexive, symmetric and transitive.
Let Z be the set of integers. Show that the relation
R = {(a, b) : a, b ∈ Z and a + b is even}
is an equivalence relation on Z.
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 identity relation on set A = {a, b, c}.
Define a transitive relation ?
Write the smallest equivalence relation on the set A = {1, 2, 3} ?
Let R be a relation on N defined by x + 2y = 8. The domain of R is _______________ .
R is a relation from {11, 12, 13} to {8, 10, 12} defined by y = x − 3. Then, R−1 is ______________ .
If R is the largest equivalence relation on a set A and S is any relation on A, then _____________ .
If A = {a, b, c, d}, then a relation R = {(a, b), (b, a), (a, a)} on A is _____________ .
In the set Z of all integers, which of the following relation R is not an equivalence relation ?
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.
Let A = {1, 2, 3, 4}, B = {4, 5, 6}, C = {5, 6} Find (A × B) ∩ (A × C).
Let A = {1, 2, 3, 4}, B = {4, 5, 6}, C = {5, 6}. Find (A × B) ∪ (A × C).
Write the relation in the Roster form and hence find its domain and range:
R2 = `{("a", 1/"a") "/" 0 < "a" ≤ 5, "a" ∈ "N"}`
Consider the set A = {1, 2, 3} and R be the smallest equivalence relation on A, then R = ______
Consider the set A = {1, 2, 3} and the relation R = {(1, 2), (1, 3)}. R is a transitive relation.
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
Give an example of a map which is one-one but not onto
Let A = {1, 2, 3} and consider the relation R = {1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (1,3)}. Then R is ______.
Every relation which is symmetric and transitive is also reflexive.
R = {(1, 1), (2, 2), (1, 2), (2, 1), (2, 3)} be a relation on A, then R is ____________.
If f(x) = `1 - 1/"x", "then f"("f"(1/"x"))` ____________.
The relation R = {(1,1),(2,2),(3,3)} on {1,2,3} is ____________.
A relation 'R' in a set 'A' is called a universal relation, if each element of' A' is related to :-
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?
A relation \(R\) on \(A\) is reflexive when which condition holds?
Which pair of relations are called trivial relations?
What should be checked to determine whether a relation is symmetric?
