I proved this inequality ;

Which is a relation on $\mathbb{Z}$ s.t a and b belongs to $\mathbb{Z}$

$$a^2 - b^2 \le 7$$

is reflexive , I'm stuck at the symmetry of this relation, can anyone help? Thank you so much!


closed as unclear what you're asking by Martin R, Graham Kemp, Yanior Weg, José Carlos Santos, Shailesh May 18 at 3:13

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  • 5
    $\begingroup$ Your question is unreadable. What is the relation? Who is Abraham? $\endgroup$ – Matthew Leingang May 16 at 10:04
  • $\begingroup$ Are you sure this defines an equivalence relation? $\endgroup$ – P-addict May 16 at 10:08
  • $\begingroup$ I'm so sorry , I just edited it $\endgroup$ – lasan manujitha May 17 at 11:28
  • $\begingroup$ No we have to check and see if it is , thank you @P-addict $\endgroup$ – lasan manujitha May 17 at 11:30

The relation is not symmetric. For example we have for $a=0$ and $b=3$:

$a^2-b^2 =-9 \le 7$, but $b^2-a^2=9 > 7.$

  • $\begingroup$ yes, thank you sir , but how can we prove that for all Z ?, without giving an example $\endgroup$ – lasan manujitha May 17 at 11:29
  • $\begingroup$ He proved that it is NOT an equivalence relation. One counterexample is enough, you don't need to prove it for all $a, b \in \mathbb{Z} $. $\endgroup$ – cavallo rosso May 17 at 11:53
  • $\begingroup$ oh thank you so much @cavallorosso $\endgroup$ – lasan manujitha May 17 at 15:51

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