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This exercise comes from A First Look At Rigorous Probability (Exercise 2.7.19):

Let $\Omega$ be a finite non-empty set, and let $\mathcal{J}$ consist of all singletons in $\Omega$, together with $\emptyset$ and $\Omega$. Show that $\mathcal{J}$ is a semialgebra. Definition is semialgebra is here.

I don't think this is true. Consider the following

Suppose $\Omega = \{ \{1,2,3,4\}, 1, 2\}$. Then $\mathcal{J} = \{ \Omega, \emptyset, \{1\}, \{2\}\}$.

$\Omega \setminus \{1\} = \{ \{1,2,3,4\}, 2\}$, which cannot be written as a disjoint union of the elements in $\mathcal{J}$.

What's wrong with what I did?

Andrés E. Caicedo
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user1691278
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1 Answers1

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Note that a singleton with respect to $\Omega$ is every set that contains a single element $w \in \Omega$. So in your example, $\{1,2,3,4\}$ is actually a singleton, and hence, it belongs to $\mathcal J$.

You can prove that it is indeed a semi-álgebra. First, let $A,B \in \Omega$ , the it is clear that $A\cap B$ is either a singleton or empty, hence, $A\cap B \in \mathcal J$.

Secondly, let $A, B \in \mathcal J$, then it’s clear that $A-B$ is either empty, a singleton or the union of singletons. We then conclude that $\mathcal J$ is a semi-algebra.

Davi Barreira
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