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.
Advertisements
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
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}.
Given an example of a relation. Which is Transitive but neither reflexive nor symmetric.
Three relations R1, R2 and R3 are defined on a set A = {a, b, c} as follows:
R1 = {(a, a), (a, b), (a, c), (b, b), (b, c), (c, a), (c, b), (c, c)}
R2 = {(a, a)}
R3 = {(b, c)}
R4 = {(a, b), (b, c), (c, a)}.
Find whether or not each of the relations R1, R2, R3, R4 on A is (i) reflexive (ii) symmetric and (iii) transitive.
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.
The following relation is defined on the set of real numbers.
aRb if a – b > 0
Find whether relation is reflexive, symmetric or transitive.
Give an example of a relation which is reflexive and transitive but not symmetric?
Give an example of a relation which is symmetric and transitive but not reflexive?
Give an example of a relation which is transitive but neither reflexive nor symmetric?
Defines a relation on N :
x > y, x, y ∈ N
Determine the above relation is reflexive, symmetric and transitive.
Defines a relation on N :
x + y = 10, x, y∈ N
Determine the above relation is reflexive, symmetric and transitive.
Show that the relation R, defined in the set A of all polygons as R = {(P1, P2) : P1 and P2 have the same number of sides}, is an equivalence relation. What is the set of all elements in A related to the right-angled triangle T with sides 3, 4 and 5?
Let S be a relation on the set R of all real numbers defined by
S = {(a, b) ∈ R × R : a2 + b2 = 1}
Prove that S is not an equivalence relation on R.
If R and S are relations on a set A, then prove that R and S are symmetric ⇒ R ∩ S and R ∪ S are symmetric ?
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.
A = {1, 2, 3, 4, 5, 6, 7, 8} and if R = {(x, y) : y is one half of x; x, y ∈ A} is a relation on A, then write R as a set of ordered pairs.
Let A = {2, 3, 4, 5, ..., 17, 18}. Let '≃' be the equivalence relation on A × A, cartesian product of Awith itself, defined by (a, b) ≃ (c, d) if ad = bc. Then, the number of ordered pairs of the equivalence class of (3, 2) is _______________ .
If R is a relation on the set A = {1, 2, 3} given by R = {(1, 1), (2, 2), (3, 3)}, then R 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 _______________ .
If A = {a, b, c}, B = (x , y} find B × A.
Consider the set A = {1, 2, 3} and R be the smallest equivalence relation on A, then R = ______
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 the relation R be defined in N by aRb if 2a + 3b = 30. Then R = ______.
An integer m is said to be related to another integer n if m is a integral multiple of n. This relation in Z is reflexive, symmetric and transitive.
R = {(1, 1), (2, 2), (1, 2), (2, 1), (2, 3)} be a relation on A, then R is ____________.
Let A = {1, 2, 3}, then the domain of the relation R = {(1, 1), (2, 3), (2, 1)} defined on A is ____________.
A relation R in set A = {1, 2, 3} is defined as R = {(1, 1), (1, 2), (2, 2), (3, 3)}. Which of the following ordered pair in R shall be removed to make it an equivalence relation in A?
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?
A relation in a set 'A' is known as empty relation:-
A relation 'R' in a set 'A' is called reflexive, if
