# Is the following relation reflexive, symmetric and transitive?

Consider the following relation in $$\mathbb{R}$$: $$x \sim y \Leftrightarrow y^2 \leq 9-x^2$$. Is this relation reflexive, symmetric and transitive?

I know it's not reflexive because, for example, for $$x=4$$, $$x \nsim x$$.

I think it's symmetric, but I'm not sure how to prove that. And I'm unable to indentify whether this is transitive or not.

• Note that $a^2≤N-b^2\iff b^2≤N-a^2$, just from basic properties of inequalities. AS to transitivity, just try some values.
– lulu
Dec 24, 2021 at 14:53
• Note that your inequality is equivalent to $x^2 + y^2 \le 9$. Since addition is commutative, symmetry follows. Dec 24, 2021 at 14:59

The relation is symmetric, because if $$y^2 \le 9-x^2$$ then we can write $$y^2 \le 9-x^2 \Leftrightarrow y^2+x^2 \le 9 \Leftrightarrow x^2 \le 9-y^2$$
For the transitivity, we can use again that $$x \sim y \Leftrightarrow x^2+y^2 \le 9$$ to construct a counterexample: $$x=3, y=0, z=3$$. We have $$x \sim y$$ and $$y \sim z$$ but $$x \not \sim z$$.