Suppose $X,Y$ are sets, and $f:X\to Y$ and $g: Y \to X$. Then there are disjoint subsets $X_1,X_2 \subseteq X$ with $X_1\cup X_2= X$ and disjoint subsets $Y_1,Y_2 \subseteq Y X$ with $Y_1\cup Y_2= Y$ such that

• $f(X_1) = Y_1$, and
• $g(Y_2) = X_2$.

(This curious result is a consequence of the Knaster-Tarski fixed point theorem.)

Can this statement be generalized to partial functions, or even binary relations $R_1, R_2 \subseteq X\times Y$?

Suppose $X,Y$ are sets, and $f:X\to Y$ and $g: Y \to X$. Then there are disjoint subsets $X_1,X_2 \subseteq X$ with $X_1\cup X_2= X$ and disjoint subsets $Y_1,Y_2 \subseteq Y$ with $Y_1\cup Y_2= Y$ such that

• $f(X_1) = Y_1$, and
• $g(Y_2) = X_2$.

(This curious result is a consequence of the Knaster-Tarski fixed point theorem.)

Can this statement be generalized to partial functions, or even binary relations $R_1, R_2 \subseteq X\times Y$?

Suppose $X,Y$ are sets, and $f:X\to Y$ and $g: Y \to X$. Then there are disjoint sets $X_1,X_2 \subseteq X$ with $X_1\cup X_2= X$ and disjoint sets $Y_1,Y_2 \subseteq X$ with $Y_1\cup Y_2= Y$ such that

• $f(X_1) = Y_1$, and
• $g(Y_2) = X_2$.

(This curious result is a consequence of the Knaster-Tarski fixed point theorem.)

Can this statement be generalized to partial functions, or even binary relations $R_1, R_2 \subseteq X\times Y$?

Suppose $X,Y$ are sets, and $f:X\to Y$ and $g: Y \to X$. Then there are disjoint subsets $X_1,X_2 \subseteq X$ with $X_1\cup X_2= X$ and disjoint subsets $Y_1,Y_2 \subseteq Y X$ with $Y_1\cup Y_2= Y$ such that

• $f(X_1) = Y_1$, and
• $g(Y_2) = X_2$.

(This curious result is a consequence of the Knaster-Tarski fixed point theorem.)

Can this statement be generalized to partial functions, or even binary relations $R_1, R_2 \subseteq X\times Y$?