Question 27

Let $A = \{1, 2, 3\}$, with $|A| = 3$. The Cartesian product $A \times A$ contains $|A|^2 = 3^2 = 9$ ordered pairs.

Total number of binary relations:

• A binary relation on $A$ is any subset of $A \times A$.
• The total number of relations is $|R| = 2^9 = 512$.

Number of reflexive relations:

• A relation is reflexive if and only if it contains all 3 diagonal pairs: $(1, 1)$, $(2, 2)$, and $(3, 3)$.
• For each of the remaining $9 - 3 = 6$ non-diagonal pairs, there are 2 choices (include or exclude).
• Total number of reflexive relations is $2^6 = 64$.

Probability of choosing a reflexive relation:

Rounding to 3 decimal places gives 0.125.