Question

Show that the relation R=∅ on a nonempty set S is symmetric and transitive, but not reflexive.

Answer 1

A binary relation $R$ is called reflexive if $(a,a)\in R$ for any $a\in S.$ Since $S$ is a nonempty set, there is an element $s\in S.$ Taking into account that $R=\emptyset$, we conclude that $(s,s)\notin R$ and thus $R$ is not reflexive.

A binary relation $R$ on a set $S$ is called symmetric if $(a,b)\in R$ implies $(b,a)\in R$. Since $R=\emptyset$, 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 $(a,c)\in R$. Since $R=\emptyset$, 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.