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I have a bit of a thinking dilemma here...

The property is (in case of a union): $A \cup(B\cap C) = (A \cup B) \cap (A \cup C)$.

To show this, let us simplify the notation in the following way: $A$ means that $x \in A$.

Therefore the LHS of the equation tells us:

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

The next line of my proof says that the above is equivalent to:

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

And I have brain damage in the above (I can easily see why this is the case if I draw the Venn diagram, but I want to understand this way of thinking too). To me the above statement means $4$ things, namely:

1.$A \ \ \text{and} \ \ A$ in line with my interpretation of the LHS

2.$A \ \ \text{and} \ \ C$ not in line with my interpretation of the LHS

3.$B \ \ \text{and} \ \ A$ not in line with my interpretation of the LHS

4.$B \ \ \text{and} \ \ C$ in line with my interpretation of the LHS

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    Have you studied propositional logic? Because $x\in A\cup(B\cap C)$ means $$x \in A\cup (B\cap C) \text{ means } x \in A \lor (x \in B \land x\in C) \equiv (x\in A \lor x\in B) \land (x\in A \lor x\in C) $$ which is equivalent to $x \in (A\cup B )\cap (A\cup C)$. – 2017-02-01
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    I have not studied propositional logic – 2017-02-01
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    Well, the process above is what we call "element chasing". or = $\lor$, and = $\land$ – 2017-02-01

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