Let $A$, $B$ be fixed sets.

What "means" the formula $Y \cap \alpha X \neq \emptyset \Leftrightarrow X \cap \beta Y \neq \emptyset$ for functions $\alpha:\mathscr{P}A\rightarrow\mathscr{P}B$ and $\beta:\mathscr{P}B\rightarrow\mathscr{P}A$? That is, how such pairs $(\alpha;\beta)$ can be represented?

Particularly, do there necessarily exist binary relations $p$ and $q$ such that $\alpha X=p[X]$ and $\beta Y=q[Y]$ for every $X\in\mathscr{P}A$, $Y\in\mathscr{P}B$?

By definition $y\in p[X] \Leftrightarrow \exists x\in X: (x,y)\in p$.


Such binary relations $p$ and $q$ exist. The proof:

$Y \cap \alpha X \neq \emptyset \Leftrightarrow X \cap \beta Y \neq \emptyset$;

$y \in \alpha X \Leftrightarrow X \cap \beta \{ y \} \neq \emptyset$;

$y \in \alpha X \Leftrightarrow \exists x \in X : x \in \beta \{ y \}$;

$y \in \alpha X \Leftrightarrow \exists x \in X : (x, y) \in p \Leftrightarrow y \in p [X]$ where $(x, y) \in p \Leftrightarrow x \in \beta \{ y \}$.

So $\alpha X = p [X]$. Similarly $\beta Y = q [Y]$ for a binary relation $q$ ($(y ; x) \in q \Leftrightarrow y \in \alpha \{ x \}$).

Also $(x, y) \in p \Leftrightarrow x \in \beta \{ y \} \Leftrightarrow \{ x \} \cap \beta \{ y \} \neq \emptyset \Leftrightarrow \{ y \} \cap \alpha \{ x \} \neq \emptyset \Leftrightarrow y \in \alpha \{ x \} \Leftrightarrow (y, x) \in q$. So $q = p^{- 1}$.

Also the formula $Y \cap \alpha X \neq \emptyset \Leftrightarrow X \cap \beta Y \neq \emptyset$ holds if $\alpha X=p[X]$ and $\beta Y=q[Y]$.

So this pair $(\alpha,\beta)$ is essentially just a binary relation $p=q^{-1}$.


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.