Define a relation \(R\) on \(A=\{1,2,3,4\}\) as \(x R y\) if \(x\) divides \(y . R\) is

Given set \(A=\{1,2,3,4\}\) and relation, \(x R y\) if \(x\) divides \(y\).
\(\Rightarrow\) Relation
\(=\{(1,1),(2,2),(3,3),(4,4),(1,2),(1,3),\), \((1,4),(2,4)\}\)
Reflexive We have, \(x R y \Leftrightarrow y / x\) for \(x, y \in A\)
For any \(x \in A\), we have \(x / x \Rightarrow x R x\).
Thus, \(x R x\) for all \(x \in A\). So, \(R\) is reflexive on \(A\).
Symmetry \(R\) ia not symmetry because, if \(y / x\), then \(x\) may not divide \(y\). For example \(4 / 2\) but \(2 / 4\).
Transitive, Let \(x, y, z \in A\), such that \(x R y\) and yRz.
Then, \(x R y\) and \(y R z \Rightarrow \frac{y}{x}\) and \(\frac{z}{y} \Rightarrow \frac{z}{x}\). So, \(R\) is a transitive relation on \(A\).