Relation R on the set A= 1,2, ....... 15 defined as R = (x, y): y- 4x…

The correct answer is Option (2) → Neither Reflexive nor symmetric

Relation $R$ on $A=\{1,2,\ldots,15\}$ is defined by:

$R=\{(x,y)\;:\; y-4x=0\}$

So $y=4x$.

Check reflexive: For reflexivity, each element must satisfy $(x,x)\in R$.

That requires $x=4x\;\Rightarrow\;3x=0$ which is impossible for any $x\in A$.

So $R$ is not reflexive.

Check symmetric: If $(x,y)\in R$, then $y=4x$. For symmetry, $(y,x)$ should satisfy $x=4y$.

But $x=4y=16x$ gives $15x=0$, impossible for $x\in A$.

Thus symmetry fails.

Hence $R$ is:

Neither reflexive nor symmetric.