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Given functions $$f_1 : A\to B$$and$$f_2 : A\to B,$$ let us write $f_1 \equiv f_2$ when there exist bijections $\alpha : A\to A$ and $\beta : B \to B$ such that $f_2(\alpha(a)) = \beta(f_1(a))$ for all $a\in A_1$.

(a) Show that $\equiv$ is an equivalence relation.

(b)How many equivalence classes of functions $f : A\to B$ such that $|A| = |B| = 3$ are there?

I tried to prove reflexivity, symmetry and transitivity.

5xum
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Giorgia
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1 Answers1

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Hints:

  1. Proving reflexivity is very easy. Try using the simplest bijections $\alpha$ and $\beta$ you can think of.

  2. For symmetry, try playing around with $\alpha^{-1}$ and $\beta^{-1}$.

  3. For transitivity, write down what $f_1\equiv f_2$ and $f_2\equiv f_3$ means.

5xum
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  • For reflexivity I have to show that f(α(a)) = β(f(a). But I don't understand, how can I use the bijectivity of α and β to prove it. – Giorgia May 27 '14 at 09:08
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    No, you have to find some bijections $\alpha$ and $\beta$ such that $f(\alpha(a))=\beta(f(a))$. – 5xum May 27 '14 at 09:12