One of the most important relations in modern mathematics are equivalence relations. For this important properties must be shown. Relationship Properties We start the section with the properties of a relation $\mathcal{R}$ on a set $A$. As a starting point, here is the definition: Definition Let $\mathcal{R}$ be a relation on a set $A.$ Then we say that: $\mathcal{R}$ is reflexive if the condition: $(\forall x \in A)(x\mathcal{R} x)$ is true if for all $x \in A, (x, x) \in R.$ $\mathcal{R}$ is symmetric if the condition: $(\forall x,y \in A)(x\mathcal{R} y \longrightarrow y\mathcal{R} x)$ is true. if for all $x, y \in A,$ if $(x, y) \in \mathcal{R} ,$ then $(y, x) \in \mathcal{R} .$ $\mathcal{R}$ is transitive if the condition: $(\forall x, y, z \in A)(x\mathcal{R} y \wedge y\mathcal{R} z \longrightarrow x\mathcal{R} z)$ is true if for all $x, y, z \in A,$ if $(x, y) \in \mathcal{R} $ and $(y, z) \in \mathcal{R},$ then $(x, z) \in \mathcal{R}.$ $\mathcal{R}$ is anti-symmetric if the condition: $(\forall x, y \in A)(x\mathcal{R} y \wedge y\mathcal{R} x \longrightarrow x = y)$ is true if for all $x, y \in A,$ if $(x, y) \in \mathcal{R}$ and $(y, x) ∈ \mathcal{R},$ then $x = y.$ Examples Below we present some examples of relationship properties: