Advertisements
Advertisements
Question
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.
Advertisements
Solution
We have,
A = {1, 2, 3} and R = {(1, 2), (1, 1), (2, 3)}
To make R a transitive relation on A, (1, 3) must be added to it.
So, the minimum number of ordered pairs that may be added to R to make it a transitive relation is 1.
APPEARS IN
RELATED QUESTIONS
If R=[(x, y) : x+2y=8] is a relation on N, write the range of R.
Show that the relation R in R defined as R = {(a, b) : a ≤ b}, is reflexive and transitive but not symmetric.
Show that the relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} is symmetric but neither reflexive nor transitive.
Let L be the set of all lines in the 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.
Let R be the relation in the set N given by R = {(a, b) : a = b – 2, b > 6}. Choose the correct answer.
Let A = {1, 2, 3}. Then number of relations containing (1, 2) and (1, 3) which are reflexive and symmetric but not transitive is ______.
Let A = {x ∈ Z : 0 ≤ x ≤ 12}. Show that R = {(a, b) : a, b ∈ A, |a – b| is divisible by 4}is an equivalence relation. Find the set of all elements related to 1. Also write the equivalence class [2]
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}
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.
If R = {(x, y) : x2 + y2 ≤ 4; x, y ∈ Z} is a relation on Z, write the domain of R.
Let R be a relation on the set N given by
R = {(a, b) : a = b − 2, b > 6}. Then,
Let R be a relation on N defined by x + 2y = 8. The domain of R is _______________ .
Let A = {1, 2, 3}. Then, the number of equivalence relations containing (1, 2) is ______.
Mark the correct alternative in the following question:
Let R be a relation on the set N of natural numbers defined by nRm if n divides m. Then, R is _____________ .
If A = {a, b, c}, B = (x , y} find A × B.
If A = {a, b, c}, B = (x , y} find B × B.
Let R be a relation on the set N of natural numbers defined by nRm if n divides m. Then R is ______.
Consider the set A = {1, 2, 3} and R be the smallest equivalence relation on A, then R = ______
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
Let R be relation defined on the set of natural number N as follows:
R = {(x, y): x ∈N, y ∈N, 2x + y = 41}. Find the domain and range of the relation R. Also verify whether R is 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
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 the relation R be defined on the set A = {1, 2, 3, 4, 5} by R = {(a, b) : |a2 – b2| < 8. Then R is given by ______.
Let A = {1, 2, 3}, then the relation R = {(1, 1), (1, 2), (2, 1)} on A is ____________.
Let A = {1, 2, 3, 4, 5, 6} Which of the following partitions of A correspond to an equivalence relation on A?
A relation R on a non – empty set A is an equivalence relation if it 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 R be a relation on the set N of natural numbers denoted by nRm ⇔ n is a factor of m (i.e. n | m). Then, R is ____________.
Given set A = {a, b, c}. An identity relation in set A 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 ____________.
An organization conducted a bike race under 2 different categories-boys and girls. Totally there were 250 participants. Among all of them finally, three from Category 1 and two from Category 2 were selected for the final race. Ravi forms two sets B and G with these participants for his college project. Let B = {b1,b2,b3} G={g1,g2} where B represents the set of boys selected and G the set of girls who were selected for the final race.
Ravi decides to explore these sets for various types of relations and functions.
- Ravi wishes to form all the relations possible from B to G. How many such relations are possible?
An organization conducted a bike race under 2 different categories-boys and girls. Totally there were 250 participants. Among all of them finally, three from Category 1 and two from Category 2 were selected for the final race. Ravi forms two sets B and G with these participants for his college project. Let B = {b1,b2,b3} G={g1,g2} where B represents the set of boys selected and G the set of girls who were selected for the final race.
Ravi decides to explore these sets for various types of relations and functions.
- Let R: B → B be defined by R = {(x, y): x and y are students of same sex}, Then this 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
The number of surjective functions from A to B where A = {1, 2, 3, 4} and B = {a, b} is
A relation 'R' in a set 'A' is called a universal relation, if each element of' A' is related to :-
Let R = {(a, b): a = a2} for all, a, b ∈ N, then R salifies.
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 ______.
