Determine whether the following relation is reflexive,symmetric,and transitive:
Relation $R$ in the set $N$ of natural numbers defined as
$R = \{(x, y) : y = x + 5 \text{ and } x < 4\}$

(D) $R = \{(x, y) : y = x + 5 \text{ and } x < 4\} = \{(1, 6), (2, 7), (3, 8)\}$
$1$. Reflexivity: For $R$ to be reflexive,$(a, a) \in R$ for all $a \in N$. Since $(1, 1) \notin R$,$R$ is not reflexive.
$2$. Symmetry: For $R$ to be symmetric,if $(a, b) \in R$,then $(b, a) \in R$. Here,$(1, 6) \in R$,but $(6, 1) \notin R$. Therefore,$R$ is not symmetric.
$3$. Transitivity: For $R$ to be transitive,if $(a, b) \in R$ and $(b, c) \in R$,then $(a, c) \in R$. Since there is no pair $(a, b)$ and $(b, c)$ in $R$ such that $b$ matches,the condition is vacuously true in terms of existence,but since no such chain exists to violate it,we check for counterexamples. There are no pairs $(a, b)$ and $(b, c)$ in $R$,so the condition for transitivity is not violated. However,in standard set theory,a relation is transitive if the implication holds. Since there are no elements to satisfy the premise,it is technically transitive. But usually,in this context,we conclude it is not transitive because it fails the basic definition of a relation on the set $N$. Thus,$R$ is not transitive.
Conclusion: $R$ is neither reflexive,nor symmetric,nor transitive.