Determine whether the following relation $R$ in the set $A = \{1, 2, 3, 4, 5, 6\}$ defined by $R = \{(x, y) : y \text{ is divisible by } x\}$ is reflexive,symmetric,and transitive.

(N/A) $A = \{1, 2, 3, 4, 5, 6\}$
$R = \{(x, y) : y \text{ is divisible by } x\}$
$1.$ Reflexivity:
Since any number $x$ is divisible by itself,$(x, x) \in R$ for all $x \in A$.
Therefore,$R$ is reflexive.
$2.$ Symmetry:
We have $(2, 4) \in R$ because $4$ is divisible by $2$.
However,$(4, 2) \notin R$ because $2$ is not divisible by $4$.
Therefore,$R$ is not symmetric.
$3.$ Transitivity:
Let $(x, y) \in R$ and $(y, z) \in R$. This means $y$ is divisible by $x$ and $z$ is divisible by $y$.
If $y = kx$ and $z = my$ for some integers $k, m$,then $z = m(kx) = (mk)x$.
Thus,$z$ is divisible by $x$,so $(x, z) \in R$.
Therefore,$R$ is transitive.
Conclusion: $R$ is reflexive and transitive but not symmetric.