Gate CS 2020 | Question - 17 | Discrete Mathematics

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:

0.125

Solution:

Sample space = 2⁹ (no. of relations on set A with n elements = $2^{n^{2}}$ , here n = 3)

For n-reflexive pairs there are (n² - n) non-reflexive pairs.

No. of reflexive relations = $2^{(n^{2} - n)}$

The Probability (chosen relations is reflexive) can be given as

$= \frac{T o t a l n o . r e f l e x i v e r e l a t i o n}{t o t a l n o . o f r e l a t i o n s} = \frac{2^{(n^{2} - n)}}{2^{n^{2}}}$

$= \frac{2^{6}}{2^{9}} = \frac{1}{2^{3}} = 0.125$

Gate CS 2020 | Question - 17 | Discrete Mathematics

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