Advertisements
Advertisements
प्रश्न
If R is a relation on the set A = {1, 2, 3} given by R = {(1, 1), (2, 2), (3, 3)}, then R is ____________ .
पर्याय
reflexive
symmetric
transitive
all the three options
Advertisements
उत्तर
all the three options
R={ (a, b) : a=b and a, b ∈ A }
Reflexivity: Let a ∈ A. Then,
a = a
⇒ (a, a) ∈ R for all a ∈ A
So, R is reflexive on A.
Symmetry : Let a, b ∈ A such that (a, b) ∈ R. Then,
(a, b) ∈ R
⇒ a = b
⇒ b = a
⇒ (b, a)∈ R for all a ∈ A
So, R is symmetric on A.
Transitivity : Let a, b, c ∈ A such that (a, b) ∈ R and (b, c) ∈ R. Then,
(a, b) ∈ R ⇒ a = b
and (b, c) ∈ R ⇒ b = c
⇒ a = c
⇒ (a, c)∈ R for all a ∈ A
So, R is transitive on A.
Hence, R is an equivalence relation on A.
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.
Show that the relation R in the set R of real numbers, defined as R = {(a, b) : a ≤ b2} is neither reflexive nor symmetric nor transitive.
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 A = {1, 2, 3}. Then number of relations containing (1, 2) and (1, 3) which are reflexive and symmetric but not transitive is ______.
Test whether the following relation R1 is (i) reflexive (ii) symmetric and (iii) transitive :
R1 on Q0 defined by (a, b) ∈ R1 ⇔ a = 1/b.
The following relation is defined on the set of real numbers. aRb if |a| ≤ b
Find whether relation is reflexive, symmetric or transitive.
Give an example of a relation which is reflexive and transitive but not symmetric?
Let A = {a, b, c} and the relation R be defined on A as follows: R = {(a, a), (b, c), (a, b)}. Then, write minimum number of ordered pairs to be added in R to make it reflexive and transitive.
Let n be a fixed positive integer. Define a relation R on Z as follows:
(a, b) ∈ R ⇔ a − b is divisible by n.
Show that R is an equivalence relation on Z.
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.
Let O be the origin. We define a relation between two points P and Q in a plane if OP = OQ. Show that the relation, so defined is an equivalence relation.
Let R be the relation defined on 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}.
Let A = {3, 5, 7}, B = {2, 6, 10} and R be a relation from A to B defined by R = {(x, y) : x and y are relatively prime}. Then, write R and R−1.
Let R = {(a, a3) : a is a prime number less than 5} be a relation. Find the range of R.
If A = {1, 2, 3}, B = {1, 4, 6, 9} and R is a relation from A to B defined by 'x is greater than y'. The range 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 A = {a, b, c, d}, then a relation R = {(a, b), (b, a), (a, a)} on A is _____________ .
Let R be the relation on the set A = {1, 2, 3, 4} given by R = {(1, 2), (2, 2), (1, 1), (4, 4), (1, 3), (3, 3), (3, 2)}. Then, _____________________ .
S is a relation over the set R of all real numbers and it is given by (a, b) ∈ S ⇔ ab ≥ 0. Then, S is _______________ .
Mark the correct alternative in the following question:
The relation S defined on the set R of all real number by the rule aSb if a b is _______________ .
For the matrix A = `[(2,3),(5,7)]`, find (A + A') and verify that it is a symmetric matrix.
Write the relation in the Roster form and hence find its domain and range :
R1 = {(a, a2) / a is prime number less than 15}
Let R be a relation on the set N of natural numbers defined by nRm if n divides m. Then R is ______.
Let L denote the set of all straight lines in a plane. Let a relation R be defined by lRm if and only if l is perpendicular to m ∀ l, m ∈ L. Then R is ______.
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
Given A = {2, 3, 4}, B = {2, 5, 6, 7}. Construct an example of the following:
a mapping from B to A
The following defines a relation on N:
x + y = 10, x, y ∈ N
Determine which of the above relations are reflexive, symmetric and transitive.
Consider the 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 ______.
Let A = { 2, 3, 6 } Which of the following relations on A are reflexive?
A relation R on a non – empty set A is an equivalence relation if it is ____________.
Let A = {1, 2, 3, …. n} and B = {a, b}. Then the number of surjections from A into B is ____________.
Let R be the relation “is congruent to” on the set of all triangles in a plane is ____________.
Given set A = {a, b, c}. An identity relation in set A is ____________.
If A is a finite set consisting of n elements, then the number of reflexive relations on A is
A market research group conducted a survey of 2000 consumers and reported that 1720 consumers like product P1 and 1450 consumers like product P2. What is the least number that must have liked both the products?
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 ______.
Let L be a set of all straight lines in a plane. The relation R on L defined as 'perpendicular to' is ______.
If a relation R on the set {a, b, c} defined by R = {(b, b)}, then classify the relation.
