I have a question in regard to some questions I am working on in beginner set theory.
I will give an example to illustrate by question better,
For example, say we were wanting to prove
$$A \cup(B \cap C)=(A \cup B) \cap (A \cup C)$$
I know that to do this I must show that they are both subsets of each other.
But my question is mostly about the operations that are allowed,
say we are first wanting to show
$$A \cup(B \cap C) \subset (A \cup B) \cap (A \cup C)$$
Now, intuitively and from previous basic proofs I know that
$B \cap C \subset B$
and $B \cap C \subset C$
Now, here is my real question, can we treat these type of problems and proofs and unions/intersections as something like a linear operator, that is , would it always be valid to say from
$$B \cap C \subset B$$ that $$A \cup (B \cap C) \subset A \cup B$$
Ie, that we just added in a A union on both sides, almost like in a basic equation with additions and subtractions. But I am not sure if this works like this? Can anyone shed some info on that?