# To check that if this inequality is an equivalence relation on $\mathbb{Z}$ [closed]

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, ShaileshMay 18 at 3:13

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

## 1 Answer

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

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

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