Is this relation symmetric and transitive?

Question

Set A is given as $A = \{1,2,3,4,5,6,7,8,9,10,11,13,14\} $ And is defined as $R = \{(x,y) : 3x = y\}$

The relation that I'm getting is: $ R = \{(3,3), (6,6), (9,9), (12,12)\} $

Over here, it is clearly visible that it is symmetric but is it transitive also?

However, the main problem is that the 'answer' says that it neither symmetric nor transitive! Can someone please clear the confusion here.

Answer 1

The relation will be transitive if $xRy$ and $yRz$ imply $xRz$ for all $x,y,z \in A$.

In the language of your relation, the implication you would need to prove is

$y=3x$ and $z=3y \overset{?}{\implies} z=3x$

(If you wanted to prove this, you could try substituting equations.)

Now, to prove that it is not transitive, you just need to find some $x,y,z$ in $A$ such that $y=3x$ and $z=3y$, but $z \neq 3x$.