> See also: > - Reference # Relations A **relation** is a structure which represents the *relationship* between elements of sets ## Properties of Relations when considering a number paired with itself (reflected in a mirror) > [!summary] **Reflexive vs Irreflexive** > A relation $R$ on a set $A$ is called **reflexive** if $(a,a) \in R$ for every element $a \in A$. > > --- > > A relation $R$ on the set $A$ is **irreflexive** if for every $a \in A$, $(a,a) \notin R$. That is, $R$ is irreflexive if no element in $A$ is related to itself Not being one does not imply the other A relation over the empty set is both reflexive and irreflexive - trivially true: - vacuously true: no way to form a contradiction --- > [!summary] **Symmetric Relations** > aa > [!summary] **Anti-Symmetric Relations** > aa --- > [!summary]+ **Transitive Relations** > A relation $R$ on set $A$ is called **transitive** if whenever $(a,b) \in R$ and $(b,c) \in R$, then $(a, c) \in R$, for all $a,b,c \in A$. In some relations, an element is always related to itself > [!summary]+ **Complete Relations** > ## Ordering all orders are antisymmetric