Advertisements
Advertisements
प्रश्न
Show that the relation R on the set Z of integers, given by R = {(a,b):2divides (a - b)} is an equivalence relation.
उत्तर
The relation R on Z is given by R = {(a,b) :2divides a - b}.
We observe the following properties of relation R.
Refelxivity : For any a ∈ Z
a - a = 0 = 0 × 2
⇒ 2 divides a - a
⇒ (a, a) ∈ R
So, R is a reflexive relation on Z.
Symmetry: Let a,b ∈ Z be such that
(a,b) ∈ R
⇒ 2 divides a - b
⇒ a - b = 2λ for some λ ∈ Z
⇒ b - a = 2(- λ ),where - λ ∈ Z
⇒ 2 divides b - a
⇒ (b, a) ∈ R
Thus, (a,b) ∈ R ⇒ (b, a) ∈ R. So, R is a symmetric relation on Z.
Transitivity: Let a,b, c ∈ Z be such that (a,b) ∈ R and (b, c) ∈ R. Then,
(a,b) ∈ R ⇒ 2 divides a - b ⇒ a - b = 2λ for some λ ∈ Z
and (b, c) ∈ R ⇒ 2 divides b - c ⇒ b - c = 2 μ for some μ ∈ Z
a - b + b - c = 2( λ + μ )
2 divides a - c
⇒ (a, c) ∈ R
Thus, (a,b) ∈ R and (b, c) ∈ R ⇒ (a, c) ∈ R.
So, R is a transitive relation on Z.
Since R is symmetric and transitive
reflexive therefore an equivalence relation
Hence, R is a transitive relation on Z.
APPEARS IN
संबंधित प्रश्न
If R=[(x, y) : x+2y=8] is a relation on N, write the range of R.
determination of whether the following relations are reflexive, symmetric, and transitive:
Relation R in the set A = {1, 2, 3...13, 14} defined as R = {(x,y):3x - y = 0}
determination of whether the following relations are reflexive, symmetric, and transitive:
Relation R in the set N of natural numbers defined as
R = {(x, y): y = x + 5 and x < 4}
determination of whether the following relations are reflexive, symmetric, and transitive:
Relation R in the set Z of all integers defined as
R = {(x, y): x − y is an integer}
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.
Given the relation R = {(1, 2), (2, 3)} on the set A = {1, 2, 3}, add a minimum number of ordered pairs so that the enlarged relation is symmeteric, transitive and reflexive.
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.
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.
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}.
If R is a symmetric relation on a set A, then write a relation between R and R−1.
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.
Let the relation R be defined on N by aRb iff 2a + 3b = 30. Then write R as a set of ordered pairs
R is a relation on the set Z of integers and it is given by
(x, y) ∈ R ⇔ | x − y | ≤ 1. Then, R is ______________ .
If A = {a, b, c}, then the relation R = {(b, c)} on A is _______________ .
If A = {1, 2, 3}, then a relation R = {(2, 3)} on A is _____________ .
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:
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 _______________ .
Mark the correct alternative in the following question:
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 for all a, b T. Then, R is ____________ .
If A = {a, b, c}, B = (x , y} find A × B.
If A = {a, b, c}, B = (x , y} find B × B.
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 ______.
If A = {1, 2, 3, 4 }, define relations on A which have properties of being:
symmetric but neither reflexive nor transitive
Give an example of a map which is neither one-one nor onto
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.
The relation R on the set A = {1, 2, 3} defined as R = {{1, 1), (1, 2), (2, 1), (3, 3)} is reflexive, symmetric and transitive.
Let A = {1, 2, 3}. Which of the following is not an equivalence relation on A?
Total number of equivalence relations defined in the set S = {a, b, c} is ____________.
The relation > (greater than) on the set of real numbers is
Let R1 and R2 be two relations defined as follows :
R1 = {(a, b) ∈ R2 : a2 + b2 ∈ Q} and
R2 = {(a, b) ∈ R2 : a2 + b2 ∉ Q}, where Q is the set of all rational numbers. Then ______
