Advertisements
Advertisements
प्रश्न
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 ___________ .
विकल्प
reflexive
symmetric
transitive
none of these
Advertisements
उत्तर
We have,
R = {`(x, y) : x−y+sqrt2` is an irrational number; x, y ∈ R}
As, `x−x+sqrt2 = sqrt2`, which is an irrational number
⇒ (x, x) ∈ R
So, R is reflexive relation
Since, (`sqrt2`, 2) ∈ R
i.e. `sqrt2−2+sqrt2=2sqrt2−2`, which is an irrational number
but `2−sqrt2+sqrt2=2`, which is a rational number
⇒ (2, `sqrt2`) ∉ R
So, R is not symmetric relation
Also, (`sqrt2`, 2) ∈ R and (2, `2sqrt2`) ∈ R
i.e. `sqrt2−2+sqrt2=2sqrt2−2`, which is an irrational number and `2−2sqrt2+sqrt2=2−sqrt2`, which is also an irrational number
But `sqrt2−2sqrt2+sqrt2=0`, which is a rational number
⇒ `(sqrt2, 2sqrt2)` ∉ R
So, R is not transitive relation
Hence, R is Reflexive.
APPEARS IN
संबंधित प्रश्न
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 transitive but not symmetric.
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 R2 is (i) reflexive (ii) symmetric and (iii) transitive:
R2 on Z defined by (a, b) ∈ R2 ⇔ |a – b| ≤ 5
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.
Is it true that every relation which is symmetric and transitive is also reflexive? Give reasons.
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?
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 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 Z be the set of all integers and Z0 be the set of all non-zero integers. Let a relation R on Z × Z0be defined as (a, b) R (c, d) ⇔ ad = bc for all (a, b), (c, d) ∈ Z × Z0,
Prove that R is an equivalence relation on Z × Z0.
Write the domain of the relation R defined on the set Z of integers as follows:-
(a, b) ∈ R ⇔ a2 + b2 = 25
If R = {(x, y) : x + 2y = 8} is a relation on N by, then write the range of R.
Define an equivalence 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].
The relation R defined on the set A = {1, 2, 3, 4, 5} by
R = {(a, b) : | a2 − b2 | < 16} is given by ______________ .
A relation R is defined from {2, 3, 4, 5} to {3, 6, 7, 10} by : x R y ⇔ x is relatively prime to y. Then, domain of R is ______________ .
Let R be a relation on N defined by x + 2y = 8. The domain of R is _______________ .
If A = {a, b, c, d}, then a relation R = {(a, b), (b, a), (a, a)} on A is _____________ .
If A = {1, 2, 3}, then a relation R = {(2, 3)} on A is _____________ .
Show that the relation R on the set Z of all integers, given by R = {(a,b) : 2 divides (a-b)} is an equivalence relation.
If A = {a, b, c}, B = (x , y} find A × A.
R = {(a, b) / b = a + 1, a ∈ Z, 0 < a < 5}. Find the Range of R.
Consider the set A = {1, 2, 3} and the relation R = {(1, 2), (1, 3)}. R is a transitive relation.
Given A = {2, 3, 4}, B = {2, 5, 6, 7}. Construct an example of the following:
a mapping from A to B which is not injective
The following defines a relation on N:
x y is square of an integer 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 ______.
The relation R = {(1,1),(2,2),(3,3)} on {1,2,3} is ____________.
If A is a finite set consisting of n elements, then the number of reflexive relations on A is
There are 600 student in a school. If 400 of them can speak Telugu, 300 can speak Hindi, then the number of students who can speak both Telugu and Hindi is:
A relation in a set 'A' is known as empty relation:-
Let a set A = A1 ∪ A2 ∪ ... ∪ Ak, where Ai ∩ Aj = Φ for i ≠ j, 1 ≤ i, j ≤ k. Define the relation R from A to A by R = {(x, y): y ∈ Ai if and only if x ∈ Ai, 1 ≤ i ≤ k}. Then, R is ______.
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 N be the set of all natural numbers and R be a relation on N × N defined by (a, b) R (c, d) `⇔` ad = bc for all (a, b), (c, d) ∈ N × N. Show that R is an equivalence relation on N × N. Also, find the equivalence class of (2, 6), i.e., [(2, 6)].
Let L be a set of all straight lines in a plane. The relation R on L defined as 'perpendicular to' is ______.
Statement 1: The intersection of two equivalence relations is always an equivalence relation.
Statement 2: The Union of two equivalence relations is always an equivalence relation.
Which one of the following is correct?
Which statement about the Identity Relation is correct?
What should be checked to determine whether a relation is reflexive?
