Advertisements
Advertisements
Question
The following relation is defined on the set of real numbers. aRb if |a| ≤ b
Find whether relation is reflexive, symmetric or transitive.
Advertisements
Solution
Reflexivity :
Let a be an arbitrary element of R. Then,
a ∈ R [Since, |a|=a]
⇒ |a|≮ a
So, R is not reflexive.
Symmetry :
Let (a, b) ∈ R
⇒ |a| ≤ b
⇒ |b| ≰ a for all a, b ∈ R
⇒ (b, a) ∉ R
So, R is not symmetric.
Transitivity :
Let (a, b) ∈ R and (b, c) ∈ R
⇒ |a| ≤ b and |b| ≤ c
Multiplying the corresponding sides, we get
|a| |b| ≤ bc
⇒ |a| ≤ c
⇒ (a, c) ∈ R
Thus, R is transitive.
APPEARS IN
RELATED QUESTIONS
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 is exactly 7 cm taller than y}
Given an example of a relation. Which is reflexive and transitive but not symmetric.
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 = {1, 2, 3}. Then number of equivalence relations containing (1, 2) is ______.
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 is father of y}
The following relation is defined on the set of real numbers.
aRb if a – b > 0
Find whether relation is reflexive, symmetric or transitive.
Prove that every identity relation on a set is reflexive, but the converse is not necessarily true.
If A = {1, 2, 3, 4} define relations on A which have properties of being reflexive, symmetric and transitive ?
Give an example of a relation which is symmetric and transitive but not reflexive?
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?
Write the domain of the relation R defined on the set Z of integers as follows:-
(a, b) ∈ R ⇔ a2 + b2 = 25
For the set A = {1, 2, 3}, define a relation R on the set A as follows:
R = {(1, 1), (2, 2), (3, 3), (1, 3)}
Write the ordered pairs to be added to R to make the smallest equivalence relation.
Let the relation R be defined on N by aRb iff 2a + 3b = 30. Then write R as a set of ordered pairs
If A = {1, 2, 3}, B = {1, 4, 6, 9} and R is a relation from A to B defined by 'x is greater than y'. The range of R 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 ______________ .
The relation R = {(1, 1), (2, 2), (3, 3)} on the set {1, 2, 3} 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 _______________ .
Show that the relation R on the set Z of integers, given by R = {(a,b):2divides (a - b)} is an equivalence relation.
If A = {a, b, c}, B = (x , y} find A × B.
If A = {a, b, c}, B = (x , y} find B × B.
Let A = {1, 2, 3, 4}, B = {4, 5, 6}, C = {5, 6} Find (A × B) ∩ (A × C).
Let A = {1, 2, 3, 4}, B = {4, 5, 6}, C = {5, 6}. Find A × (B ∪ C).
Let A = {1, 2, 3, 4}, B = {4, 5, 6}, C = {5, 6}. Find (A × B) ∪ (A × 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}
If A = {1, 2, 3, 4 }, define relations on A which have properties of being:
reflexive, symmetric and transitive
Given A = {2, 3, 4}, B = {2, 5, 6, 7}. Construct an example of the following:
an injective mapping from A to B
Give an example of a map which is one-one but not onto
Give an example of a map which is not one-one but onto
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 maximum number of equivalence relations on the set A = {1, 2, 3} are ______.
Let us define a relation R in R as aRb if a ≥ b. Then R is ______.
If A is a finite set containing n distinct elements, then the number of relations on A is equal to ____________.
Let A = {1, 2, 3, 4, 5, 6} Which of the following partitions of A correspond to an equivalence relation on A?
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 ____________.
Sherlin and Danju are playing Ludo at home during Covid-19. While rolling the dice, Sherlin’s sister Raji observed and noted the possible outcomes of the throw every time belongs to set {1,2,3,4,5,6}. Let A be the set of players while B be the set of all possible outcomes.
A = {S, D}, B = {1,2,3,4,5,6}
- Let R: B → B be defined by R = {(1,1),(1,2), (2,2), (3,3), (4,4), (5,5), (6,6)}, 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)].
