Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Much of mathematics is about solving and resolving equations, most prominently algebraic equations. But is there a general theory of resolving set builder equations?

To give an example, the equation

$$Y = \{y\ |\ (\exists x,x')\ x,x' \in X \wedge y = (x,x')\}$$

defines the set $Y= X \times X$ for a given set $X$. It can be "resolved" by

$$X = \{x\ |\ (\exists y,x')\ y \in Y \wedge y =(x,x')\}$$

which gives back the underlying set $X$ for a given set of pairs $Y$.

Another example: the equation

$$Y = \{y\ |\ (\forall x)\ x \in y \rightarrow x \in X\}$$

defines the powerset $Y = P(X)$ for a given set $X$. It can be resolved by

$$X = \{x\ |\ (\exists y)\ x \in y \wedge y \in Y \}$$

which gives back the underlying set $X$ for a given powerset $Y$.

For which set theoretic formulas such an inverse formula can be construed systematically, and how?

share|improve this question
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.