Question

Let A {1, 2, 3, 4, 5}, let R be the relation defined on A by
R= {(a, b) abEA and a divades b}. Test whether Ris reflexive,
Symmetric and transitive​

Answer 1

The relation $R$ is defined on the set $A=\{1,\ 2,\ 3,\ 4,\ 5\}$ as,

$\longrightarrow R=\{(a,\ b):a,\,b\in A,\ a\mid b\}$

Hence $R$ in roster form will be,

$\begin{aligned}\longrightarrow R\ =\ \ \{&(1,\ 1),\ (1,\ 2),\ (1,\ 3),\ (1,\ 4),\ (1,\ 5),\\&(2,\ 2),\ (2,\ 4),\ (3,\ 3),\ (4,\ 4),\ (5,\ 5)\}\end{aligned}$

Here $(1,\ 1),\,(2,\ 2),\,(3,\ 3),\,(4,\ 4),\,(5,\ 5)\in R.$ Hence $R$ is reflexive.

Here $(1,\ 2)\in R$ but $(2,\ 1)\notin R.$ Hence $R$ is not symmetric.

Here, $(1,\ 2),\,(2,\ 4)\in R$ and so $(1,\ 4)\in R.$ Hence $R$ is transitive.

Therefore, $R$ is reflexive and transitive but not symmetric.