Give an example of a relation on a set $A = \{4, 6, 8\}$ which is reflexive and symmetric but not transitive.
(N/A) Let $A = \{4, 6, 8\}$.
Define a relation $R$ on $A$ as:
$R = \{(4, 4), (6, 6), (8, 8), (4, 6), (6, 4), (6, 8), (8, 6)\}$.
$1$. Reflexive: For every $a \in A$,$(a, a) \in R$. Since $(4, 4), (6, 6), (8, 8) \in R$,the relation is reflexive.
$2$. Symmetric: For all $(a, b) \in R$,$(b, a) \in R$. Here,$(4, 6) \in R$ and $(6, 4) \in R$,and $(6, 8) \in R$ and $(8, 6) \in R$. Thus,the relation is symmetric.
$3$. Not Transitive: $A$ relation is transitive if $(a, b) \in R$ and $(b, c) \in R$ implies $(a, c) \in R$. Here,$(4, 6) \in R$ and $(6, 8) \in R$,but $(4, 8) \notin R$. Therefore,the relation is not transitive.
Hence,the relation $R$ is reflexive and symmetric but not transitive.
Define a relation $R$ on $A$ as:
$R = \{(4, 4), (6, 6), (8, 8), (4, 6), (6, 4), (6, 8), (8, 6)\}$.
$1$. Reflexive: For every $a \in A$,$(a, a) \in R$. Since $(4, 4), (6, 6), (8, 8) \in R$,the relation is reflexive.
$2$. Symmetric: For all $(a, b) \in R$,$(b, a) \in R$. Here,$(4, 6) \in R$ and $(6, 4) \in R$,and $(6, 8) \in R$ and $(8, 6) \in R$. Thus,the relation is symmetric.
$3$. Not Transitive: $A$ relation is transitive if $(a, b) \in R$ and $(b, c) \in R$ implies $(a, c) \in R$. Here,$(4, 6) \in R$ and $(6, 8) \in R$,but $(4, 8) \notin R$. Therefore,the relation is not transitive.
Hence,the relation $R$ is reflexive and symmetric but not transitive.
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