Question about the definition of a semialgebra

Question

This question has been asked here

Question about definition of Semi algebra

The OP unfortunately has selected an incorrect answer and no agreed upon correct answer has been given. The most upvoted answer finishes by saying he would be interested if someone could solve the conundrum. Therefore, I am reposting the question hoping to receive a definitively correct answer.

We say that $S$ is a semialgebra of subsets of $X$ if

1. $\emptyset \in S$
2. If $P_1,P_2 \in S$, then $P_1 \cap P_2 \in S$
3. If $P\in S$, then $X \backslash P$ can be written as a finite union of sets from $S$.

But I have found the following definition 3' is used sometimes instead of 3:

3'. If $P \in S$, then $X \backslash P$  can be written as a disjoint finite union of sets from $S$.

My question

Are the definitions 3 and 3' equivalent? If so can someone show me how we can obtain 3' from the first three conditions?

Answer 1

The Statement 3 is a combination of 3' and 2: Assuming 3' is true (then X\P can be written as a disjoint Union of sets of $S$) and moreover 2 is true. Since intersections also lie in $S$, one obtains statement 3.