Question

Show that the relation R = ∅ on the empty set S = ∅ is

reflexive, symmetric, and transitive.

Answer 1

A binary relation $R$ is called reflexive if $(a,a)\in R$ for any $a\in S.$ Since $S=\emptyset$, it contains no elements. Therefore, the statement "$a\in \emptyset=S$" is false. Consequently, the implication "if $a\in S$ then $(a,a)\in R$" is true for any $a\in S.$ It follows that $R=\emptyset$ is reflexive relation on the set $S=\emptyset$.

A binary relation $R$ on a set $S$ is called symmetric if $(a,b)\in R$ implies $(b,a)\in R$. Taking into account that $R=\emptyset$, we conclude that the statement "$(a,b)\in R$" is false. Therefore, the implication "if $(a,b)\in R$ then $(b,a)\in R$" is true. So, the relation $R=\emptyset$ is symmetric.

A binary relation $R$ on a set $S$ is called transitive if $(a,b)\in R$ and $(b,c)\in R$ implies $(a,c)\in R$. Taking into account that $R=\emptyset$, we conclude that the statement "$(a,b)\in R$ and $(b,c)\in R$" is false. Therefore, the implication "if $(a,b)\in R$ and $(b,c)\in R$ then $(a,c)\in R$" is true. So, the relation $R=\emptyset$ is transitive.