The probability that a relation R from {x, y} | vedclass

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Question

(A) Let $A = \{x, y\}$. The set $A 	imes A = \{(x, x), (x, y), (y, x), (y, y)\}$.Total number of relations on $A$ is $2^{|A 	imes A|} = 2^4 = 16$.$A

Vedclass · Accepted Answer

$\frac{5}{16}$

Vedclass · Answer

(A) Let $A = \{x, y\}$. The set $A 	imes A = \{(x, x), (x, y), (y, x), (y, y)\}$.Total number of relations on $A$ is $2^{|A 	imes A|} = 2^4 = 16$.$A$ relation $R$ is symmetric and transitive if and only if it is an equivalence relation on some subset $S \subseteq A$.The possible relations that are both symmetric and transitive are:$1$. The empty relation: $\phi$$2$. The relation $\{(x, x)\}$$3$. The relation $\{(y, y)\}$$4$. The relation $\{(x, x), (y, y)\}$$5$. The relation $\{(x, x), (y, y), (x, y), (y, x)\}$There are $5$ such relations.Therefore,the probability is $\frac{5}{16}$.

The probability that a relation $R$ from $\{x, y\}$ to $\{x, y\}$ is both symmetric and transitive is equal to:

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