Advertisements
Advertisements
प्रश्न
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 _______________ .
पर्याय
an equivalence relation
reflexive, transitive but not symmetric
symmetric, transitive but not reflexive
neither transitive nor reflexive but symmetric
Advertisements
उत्तर
We have,
S = {(a, b) : a b; a, b R}
Hence, The relation S defined on the set R of all real number by the rule aSb if a b is reflexive, transitive but not symmetric.
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)].
Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as R = {(a, b) : b = a + 1} is reflexive, symmetric or transitive.
Given an example of a relation. Which is reflexive and symmetric but not 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.
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?
The binary operation *: R x R → R is defined as a *b = 2a + b Find (2 * 3)*4
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 live in the same locality}
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.
If A = {1, 2, 3, 4} define relations on A which have properties of being reflexive, transitive but not symmetric ?
Give an example of a relation which is reflexive and symmetric but not 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.
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 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.
Define a symmetric relation ?
State the reason for the relation R on the set {1, 2, 3} given by R = {(1, 2), (2, 1)} to be transitive ?
If R is a relation on the set A = {1, 2, 3} given by R = {(1, 1), (2, 2), (3, 3)}, then R is ____________ .
In the set Z of all integers, which of the following relation R is not an equivalence relation ?
Mark the correct alternative in the following question:
Let L denote the set of all straight lines in a plane. Let a relation R be defined by lRm if l is perpendicular to m for all l, m ∈ L. 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 ____________ .
Mark the correct alternative in the following question:
For real numbers x and y, define xRy if `x-y+sqrt2` is an irrational number. Then the relation R is ___________ .
Show that the relation R on the set Z of integers, given by R = {(a,b):2divides (a - b)} is an equivalence relation.
If `f(x) = (4x + 3)/(6x - 4), x ≠ 2/3`, show that fof (x) = x for all `x ≠ 2/3`. Also, find the inverse of f.
If A = {a, b, c}, B = (x , y} find A × B.
In the set of natural numbers N, define a relation R as follows: ∀ n, m ∈ N, nRm if on division by 5 each of the integers n and m leaves the remainder less than 5, i.e. one of the numbers 0, 1, 2, 3 and 4. Show that R is equivalence relation. Also, obtain the pairwise disjoint subsets determined by R
Let R be a relation on the set N of natural numbers defined by nRm if n divides m. Then R is ______.
For real numbers x and y, define xRy if and only if x – y + `sqrt(2)` is an irrational number. Then the relation R is ______.
The following defines a relation on N:
x + 4y = 10 x, y ∈ N.
Determine which of the above relations are reflexive, symmetric and transitive.
Let us define a relation R in R as aRb if a ≥ b. Then R is ______.
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.
R = {(1, 1), (2, 2), (1, 2), (2, 1), (2, 3)} be a relation on A, then R is ____________.
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 A = {1, 2, 3, …. n} and B = {a, b}. Then the number of surjections from A into B is ____________.
A general election of Lok Sabha is a gigantic exercise. About 911 million people were eligible to vote and voter turnout was about 67%, the highest ever
Let I be the set of all citizens of India who were eligible to exercise their voting right in the general election held in 2019. A relation ‘R’ is defined on I as follows:
R = {(V1, V2) ∶ V1, V2 ∈ I and both use their voting right in the general election - 2019}
- The above-defined relation R is ____________.
Students of Grade 9, planned to plant saplings along straight lines, parallel to each other to one side of the playground ensuring that they had enough play area. Let us assume that they planted one of the rows of the saplings along the line y = x − 4. Let L be the set of all lines which are parallel on the ground and R be a relation on L.
Answer the following using the above information.
- Let R = {(L1, L2 ): L1 is parallel to L2 and L1: y = x – 4} then which of the following can be taken as L2?
The relation > (greater than) on the set of real numbers is
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 ______.
If a relation R on the set {a, b, c} defined by R = {(b, b)}, then classify the relation.
