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I must prove that the ring of parts of set $X$, with the binary operations symmetric difference (as the sum) and the Intersection (as the product) is isomorphic to the ring formed by the functions of the form $f: X \rightarrow \mathbb{Z}_{2}$ under the sum and the usual product in $\mathbb{Z}_{2}$. I have been looking for functions that constitute a homomorphism and have not been able to find it, I appreciate your help.

user26857
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Isomorphicc
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1 Answers1

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Consider the function that sends each $S \in \mathcal{P}(X)$ to the map $\chi_S : X \to \mathbb Z_2$ given by $$\chi_S(x) = \begin{cases} 1 & \textrm{if } x \in S, \\[0.5mm] 0 & \textrm{if } x \notin S. \end{cases}$$

azif00
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    I don't get it. In the question you say that $F$ is the set of all functions $X \to \mathbb Z_2$ viewed as a ring with pointwise addition and multiplication, right? So, the function $S \mapsto \chi_S$ is a map from $\mathcal{P}(X)$ to $F$. I don't see the problem here :( – azif00 Oct 11 '20 at 02:43
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    @Lorole28 Note that $\chi_{R\cap S}(x) = \chi_R(x)\cdot\chi_S(x)$ and $\chi_{R\bigtriangleup S}(x) = \chi_R(x)+\chi_S(x)$ (since the codomain is $\mathbb Z_2$). – Christoph Oct 11 '20 at 07:25
  • Yes, it was my mistake. Thanks for your explanations! :3 – Isomorphicc Oct 11 '20 at 14:58