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(B) An empty relation $\phi$ on a set $A$ is defined as $\phi = \emptyset \subseteq A 	imes A$. $1$. For reflexivity,for every $a \in A$,$(a, a)$ mus

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Symmetric and Transitive

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(B) An empty relation $\phi$ on a set $A$ is defined as $\phi = \emptyset \subseteq A 	imes A$. $1$. For reflexivity,for every $a \in A$,$(a, a)$ must be in $\phi$. Since $\phi$ is empty,it contains no elements,so it is not reflexive (unless $A = \emptyset$). $2$. For symmetry,if $(a, b) \in \phi$,then $(b, a)$ must be in $\phi$. Since there are no elements in $\phi$,the condition $(a, b) \in \phi \implies (b, a) \in \phi$ is vacuously true. Thus,it is symmetric. $3$. For transitivity,if $(a, b) \in \phi$ and $(b, c) \in \phi$,then $(a, c)$ must be in $\phi$. Since there are no elements in $\phi$,the condition is vacuously true. Thus,it is transitive. Therefore,the empty relation is symmetric and transitive.

The empty relation on a set $A$ is

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