The following proof is given to show that $\mathcal P(A) \cup \mathcal P(B) \subseteq \mathcal P(A\cup B)$ ($\mathcal P$ is the power set):

Let $X \in (\mathcal{P}(A) \cup \mathcal{P}(B))$, by definition of union: $$X \in \mathcal{P}(A) \lor X \in \mathcal{P}(B)$$ $\implies$By definition of power set $$X \subseteq A \lor X \subseteq B$$ $\implies$By union transitivity$$X \subseteq A\cup B$$ $\implies$By definition of power set $$X \in \mathcal{P}(A \cup B)$$ Then $\mathcal{P}(A) \cup \mathcal{P}(B) \subseteq \mathcal{P}(A \cup B) \blacksquare$

Now, I thought I could apply the opposite logic to show that $\mathcal P(A) \cup \mathcal P(B) \supseteq \mathcal P(A\cup B)$, and thus prove that $\mathcal P(A) \cup \mathcal P(B)= \mathcal P(A\cup B)$:

Let $X \in \mathcal{P}(A \cup B)$, By definition of power set: $$X \subseteq A\cup B$$ $\implies$ By union transitivity: $$X \subseteq A \lor X \subseteq B$$ $\implies$ By definition of powerset: $$X \in \mathcal{P}(A) \lor X \in \mathcal{P}(B)$$ $\implies$ by definition of union: $$X \in (\mathcal{P}(A) \cup \mathcal{P}(B))$$ Then $\mathcal {P}(A \cup B) \subseteq \mathcal{P}(A) \cup \mathcal{P}(B) \blacksquare$

Multiple posts here prove that for any sets $A$ or $B$, if $\mathcal P(A) \cup \mathcal P(B)= \mathcal P(A\cup B)$ then either $A \subseteq B$ or $B \subseteq A$, so clearly my opposite direction proof can't be true.

What I'd like to understand is which step is based on a false assumtion/implication?


$X \subseteq A \cup B$ doesn't imply $X \subseteq A$ or $X \subseteq B$.

For example let $A=\{1\}$ and $B=\{2\}$ and $X = A \cup B$.

$X$ is neither subset of $A$ nor subset of $B$.


Your "union transitivity" step is wrong. That is, if you have $X \subset A \cup B$, then you don't necessarily have $X \subset A$ or $A \subset B$. For example, let $A =\{1\},B=\{2\},X=\{1,2\}$.


I have lot of books in my shelf: Some of them I bought and the rest are borrowed from others and never returned.

A general subset of books from my shelf need not consist exclusively of borrowed books, or of own books.


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.