2
$\begingroup$

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.

$\endgroup$
2
  • $\begingroup$ 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. $\endgroup$
    – lulu
    Commented Dec 24, 2021 at 14:53
  • $\begingroup$ Note that your inequality is equivalent to $x^2 + y^2 \le 9$. Since addition is commutative, symmetry follows. $\endgroup$
    – Aryaman Maithani
    Commented Dec 24, 2021 at 14:59

1 Answer 1

Reset to default
3
$\begingroup$

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 $x \not \sim z$.

$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .