User:KBlott/Definition/In

Preamble
Let $(B_{\bot}^{\top}, \vee, \wedge, -, \bot, \top)$ be a Boolean algebra. Let $(U_Z^I, \cup, \cap, -, Z, I)$ be a universe algebra. Let $C \in $$U^I$ be a class in the given universe. Let $x \in $$U_Z$ be an object in that universe.

Definition
Then, in ($\in$) is a mapping
 * $U_Z \times U^I \to B_{\bot}^{\top}$

such that
 * $x \in C \implies \top$.

Notice that the definition of $\in$ is circular. The symbol $\in$ is used to define itself. This technically makes the definition an axiom. Also, the definition is very abstract and requires a somewhat generalized notion of Boolean algebra and what a universe is. Note, however, the axiom it is quite elegant.

Comments
Notice that if the Boolean domain contains only two elements (ie
 * $B_{\bot}^{\top} = \{\bot, \top\}$

with
 * $\bot \neq \top)$

then there is a dual mapping
 * $-(\ni): U_Z \times U^I \to B_{\bot}^{\top}$

such that
 * $-(x \ni C) \iff x \in C$.