Question

Let R be a relation on the set { a, b, c, d }

R = { (a, b), (a, c), (a, d), (c, b), (c, d), (d, b)}.

Identify the properties satisfied on this given relation.

Answer 1

Given, $R = \{(a, b), (a, c), (a, d), (c, b), (c, d), (d, b)\}.$

$R$ is not reflexive since $(a, a) \notin R ~\forall a$.

$R$ is irreflexive since $(a, a) \notin R ~\forall a$.

$R$ is not symmetric because for all $(a, b) \in R$, $(b, a) \notin R$.

$R$ is antisymmetric since for $x \ne y$ either $(x,y) \notin R$ or $(y,x) \notin R$ for all $x,y\in\{a,b,c,d\}$.

$R$ is transitive because for $(c, d)\in R ~\&~ (d, b)\in R$, we have $(c,b)\in R$.

$R$ is asymmetric because $(a, b) \in R \implies (b, a) \notin R$ .