Reflexive, Symmetric, Transitive, And Antisymmetric Relations

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Definition (Reflexive, Symmetric, Transitive, And Antisymmetric Relations)

Let $\mathrm{R}$ be a relation on a set $A$.

• We say that $\mathrm{R}$ is reflexive if and only if for every $a\in A$, we have $(a,a)\in \mathrm{R}$.
• We say that $\mathrm{R}$ is symmetric if and only if for every $a,b\in A$, we have if $(a,b)\in \mathrm{R}$ then $(b,a)\in \mathrm{R}$.
• We say that $\mathrm{R}$ is transitive if and only if for every $a,b,c\in A$, we have if $(a,b)\in \mathrm{R}$ and $(b,c)\in \mathrm{R}$ then $(a,c)\in \mathrm{R}$.
• We say that $\mathrm{R}$ is antisymmetric if and only if for every $a,b\in A$, we have if $(a,b)\in \mathrm{R}$ and $(b,a)\in \mathrm{R}$ then $a=b$.