Advertisements
Advertisements
Question
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.
Advertisements
Solution
Let A be the set of all points in a plane such that
A={P : P is a point in the plane}
Let R be the relation such that R={(P, Q) : P, Q∈A and OP=OQ, where O is the origin}
We observe the following properties of R.
Reflexivity: Let P be an arbitrary element of R.
The distance of a point P will remain the same from the origin.
So, OP = OP
⇒ (P, P) ∈ R
So, R is reflexive on A.
Symmetry : Let (P, Q) ∈ R
⇒ OP = OQ
⇒ OQ = OP
⇒ (Q, P) ∈ R
So, R is symmetric on A.
Transitivity: Let (P, Q), (Q, R) ∈ R
⇒ OP= OQ and OQ = OR
⇒ OP= OQ = OR
⇒ OP = OR
⇒ (P, R) ∈ R
So, R is transitive on A.
Hence, R is an equivalence relation on A.
APPEARS IN
RELATED QUESTIONS
If R=[(x, y) : x+2y=8] is a relation on N, write the range of R.
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 symmetric but neither reflexive nor transitive.
Show that the relation R in the set A of points in a plane given by R = {(P, Q) : distance of the point P from the origin is the same as the distance of the point Q from the origin} is an equivalence relation. Further, show that the set of all points related to a point P ≠ (0, 0) is the circle passing through P with the origin as its centre.
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}
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.
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?
Let A = {1, 2, 3} and R = {(1, 2), (1, 1), (2, 3)} be a relation on A. What minimum number of ordered pairs may be added to R so that it may become a transitive relation on A.
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.
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.
Define a symmetric relation ?
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 the equivalence relation on the set Z of the integers given by R = { (a, b) : 2 divides a - b }.
Write the equivalence class [0].
If A = {a, b, c}, then the relation R = {(b, c)} on A is _______________ .
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 _______________ .
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 a relation on the set A = {1, 2, 3} given by R = {(1, 1), (2, 2), (3, 3)}, then R is ____________ .
For the matrix A = `[(2,3),(5,7)]`, find (A + A') and verify that it is a symmetric matrix.
Let A = {1, 2, 3, 4}, B = {4, 5, 6}, C = {5, 6}. Find A × (B ∪ C).
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 A = {0, 1, 2, 3} and define a relation R on A as follows: R = {(0, 0), (0, 1), (0, 3), (1, 0), (1, 1), (2, 2), (3, 0), (3, 3)}. Is R reflexive? symmetric? transitive?
Let Z be the set of integers and R be the relation defined in Z such that aRb if a – b is divisible by 3. Then R partitions the set Z into ______ pairwise disjoint subsets
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
The following defines a relation on N:
x + y = 10, x, y ∈ N
Determine which of the above relations are reflexive, symmetric and transitive.
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 ______.
Let the relation R be defined in N by aRb if 2a + 3b = 30. Then R = ______.
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 S = {1, 2, 3, 4, 5} and let A = S x S. Define the relation R on A as follows:
(a, b) R (c, d) iff ad = cb. Then, R is ____________.
Given set A = {1, 2, 3} and a relation R = {(1, 2), (2, 1)}, the relation R will be ____________.
Find: `int (x + 1)/((x^2 + 1)x) dx`
The number of surjective functions from A to B where A = {1, 2, 3, 4} and B = {a, b} is
Let R = {(a, b): a = a2} for all, a, b ∈ N, then R salifies.
Let f(x)= ax2 + bx + c be such that f(1) = 3, f(–2) = λ and f(3) = 4. If f(0) + f(1) + f(–2) + f(3) = 14, then λ is equal to ______.
Let A = {1, 2, 3, 4} and let R = {(2, 2), (3, 3), (4, 4), (1, 2)} be a relation on A. Then R is ______.
Let A = {3, 5}. Then number of reflexive relations on A is ______.
