Let R be the set of all binary relations on the set {1, 2, 3}. Suppose a relation is chosen from R at random. The probability that the chosen relation is reflexive (round off to 3 decimal places) is ________.Correct answer is '0.125'. Can you explain this answer?

Question

Answer

Understanding the Problem:

We are given a set R that contains all binary relations on the set {1, 2, 3}. We need to find the probability that a randomly chosen relation from R is reflexive.

Solution:

To find the probability, we first need to determine the total number of relations in the set R. Since the set {1, 2, 3} has 3 elements, there are a total of 3 * 3 = 9 possible ordered pairs in the set R.

Counting Reflexive Relations:

A relation is reflexive if it contains the ordered pairs (1, 1), (2, 2), and (3, 3). For a relation to be reflexive, we have two choices for each of these ordered pairs: include them or exclude them.

- If we include all three ordered pairs, then the relation is reflexive.
- If we include any one or two ordered pairs, then the relation is not reflexive.
- If we exclude all three ordered pairs, then the relation is not reflexive.

Calculating the Probability:

Let's calculate the number of reflexive relations:

- If we include all three ordered pairs, there is only one possible relation.
- If we exclude all three ordered pairs, there is also only one possible relation.
- If we include any one or two ordered pairs, there are no possible reflexive relations.

Therefore, there are a total of 2 reflexive relations in the set R.

The probability of choosing a reflexive relation is given by the ratio of the number of reflexive relations to the total number of relations:

Probability = Number of reflexive relations / Total number of relations
= 2 / 9
≈ 0.222

Rounding off to 3 decimal places, the probability that the chosen relation is reflexive is 0.222.

However, the correct answer is given as 0.125. It seems like there might be a mistake in the question or the correct answer provided. The correct probability should be 0.222 based on the given information.

Similar GATE Doubts

Let R be a relation on the set A = {1, 2, 3}, such that R = {(1,1), (2,2), (3,3), (1,2)}. Now consider the following relations.I. {(1,1), (2,2), (3,3)}II. {(1,1), (2,2)}III. {(1,1), (2,2), (3,3), (1,2)}Which of the above correctly represent(s) the smallest reflexive closure of R?

The binary relation R = {(1, 1), (2, 1), (2, 2), (2, 3), (2, 4), (3, 1), (3, 2), (3, 3), (3, 4)},on the set A = {1 , 2 , 3, 4} is

A steel part with surface area of 125 cm2 is to be chrome coaled through an electroplating process using chromium acid sulphate as an electrolyte. An increasing current is applied to the part according to the following current time relation:I = 12 + 0.2twhere, I = current (

A liquid phase irreversible reaction A→ B is carried out in an adiabatic CSTR operating under steady state conditions. The reaction is elementary and follows the first order rate law. For this reaction, the figure below shows the conversion (X

The binary relation S = f (empty set) on set A = {1,2, 3} is

Top Courses for GATE

Suggested Free Tests

Related Content

Mock Test Series

Subject-wise Courses

Other Engg Branches

Related Exams

Company