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For example, on proving the distributive law of set theory, the following constitutes as a proof

Proof : enter image description here

I am new to proof involving sets but this to me seems nothing more than replacing unions with "or" and intersections with "and", which is just a re-wording. Further more this seems to borrow some other branch of mathematics (boolean algebra specifically) entirely which is a separate development from naive set theory. It is like answering the question "why is the derivative of sin(x) equal to negative cos(x)" with some explanation from differential forms or measure theory involving the definition of differential d when limits could have sufficed - why borrow another language to explain some facts in this language?

I could have stated the law with "x is either in A or in (B and C)" is equal to "(x is in A or x is in B) and (x is in A or x is in C)", and then told to construct a proof.

Can someone enlighten me what is the importance of exchanging $\cup$ with or and $\cap$ with and and how that prove anything?

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    $\begingroup$ This is something you're going to have to get used to in set theory, where you break things down into their basic definitions, combine them to create a new definition and then ta-da, that definition actually represents something else. $\endgroup$ – Zain Patel Jul 22 '15 at 22:06
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    $\begingroup$ You cannot prove much of anything in any branch of math without understanding conjunction, disjunction, implication, and inferences thereon. But perhaps if the "proof" did not mix words and mathematical notation in such a strange way, OP would not be so confused about what it is saying. $\endgroup$ – David K Jul 22 '15 at 22:38
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Let $x \in A \cup (B \cap C)$. If $x \in A \cup (B \cap C)$ then $x$ is either in $A$ or in ($B$ and $C$).

$x \in A$ or $x \in (B \ \text{and}\ C)$

To be honest, I do not like this use of language. The phrasing "is either in" simply does not seem like good English to me. It especially does not seem like good mathematical English.

Moreover, the first line makes a statement and then appears to attempt to explain what it means. The purpose apparently is to make it clearer how the next line was obtained. But the proof does not offer explanations like that for any of its other statements. Why not? Probably because these "explanations" would be redundant and confusing, as I think this first explanation is.

Furthermore, on the second line, is $(B \text{ and } C)$ a set, or is it some sort of shorthand for distributing the "and" over the symbol $\in$? Neither of these options seems like standard mathematical writing to me. This is needless ambiguity.

I might write these lines as follows:

Let $x \in A \cup (B \cap C)$.

$x \in A$ or $x \in (B \cap C$).

Then the next line follows quite easily.

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$x \in (B \text{ and } C)$ is being used as an abbreviation to "both $x \in B$ and $x \in C$ are true", e.g., the first line may be rewritten as

\begin{equation} x \in A ~ \vee (x \in B \wedge x \in C) \end{equation}

where $\vee$ is logical or and $\wedge$ is logical and.

And yes, you will need to use logic to prove things in any branch of mathematics.

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