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?

Thank you


Yes, you can prove that given $A,B,C$ sets, we have: $$B \subseteq C \implies \begin{cases} A \cup B \subseteq A \cup C \\ A \cap B \subseteq A \cap C\end{cases}$$and use it as a theorem.

Suppose that $B \subseteq C$. If $x \in A \cup B$, we have two cases: $x \in A$, and so $x \in A \cup C$, or $x \in B$, and hence $x \in C$ (by hypothesis), and we get $x \in A \cup C$ in the same way.

Suppose that $B \subseteq C$. If $x \in A \cap B$, we have $x \in A$ and $x \in B$. By hypothesis, we have $x \in A$ and $x \in C$, hence $x \in A \cap C$.

Let $X$ be the universe. Fix $A \in \wp(X)$. We can define $f_A,g_A: \wp(X) \to \wp(X)$ by: $$f_A(B) = A \cup B \quad\text{and}\quad g_A(B) = A \cap B.$$If you order $\wp(X)$ by inclusion, we can say that $f_A$ and $g_A$ are increasing maps. Above we've shown that $B \subseteq C \implies f_A(B) \subseteq f_A(C)$ and $g_A(B) \subseteq g_A(C)$.

  • $\begingroup$ Right, thanks I do understand that, my question is more about if we can always treat union/intersections as "operators", I don't know the right term though $\endgroup$ – Quality Jun 14 '15 at 22:51
  • $\begingroup$ I added a bit on that in the answer. $\endgroup$ – Ivo Terek Jun 14 '15 at 22:53

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.