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I came up with this question , which I have to prove that it is an equivalence relation

Define a function f : R → R by f(x) = x^2 + 1. For a, b ∈ R define a ≃ b to mean that f(a) = f(b)

I have done proving that it is a reflexive and also symmetric , I am bit stuck at the transitive part Thankyou so much.

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  • $\begingroup$ Suppose $ a \sim b$ , and $ b \sim c$. Then, $ f(a) = f(b)$, and $ f(b) = f(c) \implies f(a) = f(c) $. Thus, $ a \sim c$ by definition. $\endgroup$ – P-addict May 17 at 16:00
  • $\begingroup$ thank you so much @P-addict $\endgroup$ – lasan manujitha May 17 at 17:15
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If $f(a)=f(b)$ and $f(b)=f(c)$, then $f(a)=f(c)$. This works for every function, not just for that one.

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  • $\begingroup$ Thankyou so much $\endgroup$ – lasan manujitha May 17 at 17:15
  • $\begingroup$ oh I'm new , sorry , Thankyou again $\endgroup$ – lasan manujitha May 17 at 19:19

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