Suppose we define a relation $R$ in the natural set $\mathbb N$ which says: $$(x,y)\in R\iff x^2-4xy+3y^2=0$$ and we would like to find which of the following properties does $R$ satisfy.
My book gives four options:
(a) reflexive and transitive
(b) reflexive and symmetric
(c) symmetric and transitive
(d) an equivalence relation
According to me, none of the options is correct.
For example: $(9,3),(3,1)\in R$ but $(9,1)\notin R$
hence $(a,b),(b,c)\in R \not \implies (a,c)\in R$, proving $R$ to be NOT transitive.
Am I correct? Thank you.
For reflexivity, it is easy to see.
To verify whether it has symmetry, take arbitrary $x,y$ s.t. $(x,y) \in R$, then we have
$$x^2−4xy+3y^2=0$$
Can we derive $y^2 - 4xy + 3x^2 = 0$ from it? No, in general it is not true. So it doesn't have symmetry.
To verify whether it has transitivity, take $x,y, z$ s.t. $(x,y) \in R$ and $(y,z) \in R$, then we have
$$x^2−4xy+3y^2=0, y^2−4yz+3z^2=0$$
Can we derive $x^2 - 4xz + 3z^2 = 0$? Whatever we do, in general it is not true. So we don't have transitivity.
Therefore, none of the options on your book is true.