Binary Relations and its Properties

Symmetric
$$\forall a, b \bullet a R b \implies b R a$$
where \(R\) is a relation. For example,

• = relation on \(\mathbb{R}\)
• is equal to
• is married to 
• {(1,2), (2,1)}

Antisymmetric
$$\forall a,b:X \bullet a R b \land b R a \implies a = b $$
For example,

• \(\leq\)
• \(\geq\)
• \(\subseteq\)
• {(1, 2), (2, 3), (1, 1)}

Asymmetric
$$\forall a,b:X \bullet a R b \implies \lnot(bRa) $$
For example,

• {(1,2), (3,4)}

Reflexive
$$\forall x:X \bullet xRx$$
For example,

• =
• \(\leq\)

Irreflexive (strict)
$$\forall x:X \bullet \lnot(xRx)$$
For example,

• \(>\), \(<\)

Coreflexive (a subset of Identity Relation)
$$\forall x,y:X \bullet x R y \implies x= y)$$
For example,

•  equal to and odd, such as {(1,1), (3,3)}

Identity
$$\{(x,x) | x \in X\}$$

Transitive
$$\forall x,y,z:X \bullet x R y \land y R z \implies x R z$$
For example,

• \(<\), =, \(\leq\)