0
$\begingroup$

There are two statements:

$(a)\space\mathcal P(A\setminus B)\subseteq \mathcal P(A)\setminus \mathcal P(B)$

$(b)\space \mathcal P(A\setminus B)\supseteq \mathcal P(A)\setminus \mathcal P(B)$

Generally speaking: $\emptyset\in(\mathcal P(A)\cap \mathcal P(B))$

Is it correct to say power sets can be disjunctive because the empty set is disjunctive to itself? $$\emptyset\cap\emptyset=\emptyset, edited$$ My counter-example for the statement: $(a)\space\mathcal P(A\setminus B)\subseteq \mathcal P(A)\setminus \mathcal P(B)$ is a pair of two disjunctive sets $A$ and $B$ so my claim is equivalent to: $$\mathcal P(A)\nsubseteq\mathcal P(A)\setminus \{\emptyset\}$$ but is kind of a contradiction with the question above.

The counter-example for the statement:$(b)\space \mathcal P(A\setminus B)\supseteq \mathcal P(A)\setminus \mathcal P(B)$ is a pair of sets $A$ and $B$ such that: $$A\cap B\neq \{\emptyset\}$$ /this should be $\emptyset$ instead of $\{\emptyset\}$, as noticed in comments/ I also took into consideration the cardinality of the power sets, but it was insignificant when dealing with disjunctive sets. It works better for a Cartesian product.

$\endgroup$
4
  • $\begingroup$ $\{\varnothing\}$ is not the empty set. $\endgroup$
    – Asaf Karagila
    Nov 3 '19 at 11:09
  • $\begingroup$ What is a disjunctive set? $\endgroup$
    – J.-E. Pin
    Nov 3 '19 at 12:04
  • 2
    $\begingroup$ The term you're looking for is "disjoint sets". $\endgroup$
    – Asaf Karagila
    Nov 3 '19 at 12:55
  • $\begingroup$ Thank you very much. I had problems with translating from Croatian to English. $\endgroup$
    – Invisible
    Nov 3 '19 at 12:58
4
$\begingroup$

Counterexamples:

  1. The empty set is in $\mathcal P(A \setminus A)$ while $\mathcal P(A) \setminus P(A)$ is empty.
  2. The set $\{0,\pi\}$ is in $\mathcal P(R) \setminus \mathcal P(Q)$ but not in $\mathcal P(R \setminus Q)$.
$\endgroup$

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.