Test binary relation on the set for reflexivity, symmetry, antisymmetry and transitivity

Question

$$S = (-\infty,-1)$$

$$xRy, 27x^3+9xy^2 \leqslant 27x^2y+y^3$$

Can somebody help me with this, I am new in this field and I don't know how to start.

Answer 1

Welcome to the site. Can you edit your question and elaborate on some things? Like the things I mention below. You need to check the definitions of the different type of relations. For instance, for the relation to be reflexive, it has to be true that $xRx$ for all $x\in S$. What does this mean? It means that $$27x^3+9xx^2 \leq 27x^2x + x^3$$ for all $x$. Perhaps you can see if this is true or not for all $x\in S$.

As for symmetry, it needs to be true that whenever $xRy$ for $x,y\in S$, then also $yRx$, i.e. if

$$27x^3+9xy^2\leq 27x^2y+y^3,$$

is it then also true that $$27y^3+9yx^2\leq 27y^2x+x^3?$$ Anti-symmetry means that if $xRy$ and $yRx$, then we must have $x=y$. Try writing down this condition for your specific relation.

A transitive relation means that if $xRy$ and $yRz$, then it must be true that $xRz$. If you write down this condition for your specific relation, you might be able to tell if it's true or not.

Answer 2

Hint: $27x^3+9xy^2 \leqslant 27x^2y+y^3 \iff (3x)^3-y^3 \le 27x^2y-9xy^2 $.

Now compute $(3x-y)^3$ and you will see:

$$xRy \iff (3x-y)^3 \le 0 \iff 3x-y \le 0.$$