From Rosen's Discrete Mathematics and Its Applications, 3ed, chapter 9 p. 612:
How can they tell $R$ is an equivalence relation right off the bat?
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Practice. How else? – Asaf Karagila Feb 05 '20 at 09:08
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More generally, for any function $f:A\to B$, the relation $\{(a, a') :f(a) =f(a')\}$ is always an equivalence relation.
Prove it in this generality, and find the sets $A,B$ and the function $f$ to apply this for the problem.
Berci
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If I understand it correctly, they are saying $R: A\to A$ is always an equivalence relation. From the book:
A relation on a set A is called an equivalence relation if it is reflexive, symmetric, and transitive.How can they tell it is an equivalence without checking these three properties? – J. Doe Feb 05 '20 at 12:27 -
They have checked it. Check them yourself as well, in the above generality. – Berci Feb 05 '20 at 14:03