User:Atomic predicates/sandbox

Consider a set $$U$$ of elements. A predicate $$P$$ specifies a subset of elements. Predicate true specifies $$U$$. Predicate false specifies the empty set.

Definition 1 (Atomic Predicates): Given a set $$ \mathcal{P}$$ of predicates, its set of atomic predicates $$\{p_1, \dots, p_k\}$$ satisfies these five properties:
 * 1) $$ p_i\neq \text{false}, \forall i\in\{1,\dots,k\}$$.
 * 2) $$\vee_{i=1}^k p_i = \text{true}$$.
 * 3) $$p_i\wedge p_j = \text{false, if } i\neq j $$.
 * 4) Each predicate $$P\in\mathcal{P}, P\neq\text{false}$$, is equal to the disjunction of a subset of atomic predicates:
 * $$P=\bigvee_{i\in S(P)} p_i, \text{where } S(P)\subseteq \{1,\dots,k\}.$$ (1)
 * 1) $$k$$ is the minimum number such that the set $$\{p_1,\dots, p_k\}$$ satisfies the above four properties.

Note that if $$P=\text{true}$$, then $$S(P)=\{1,\dots,k\}$$; if $$P=\text{false}$$, $$S(P)=\emptyset$$. Since $$p_1,\dots,p_k$$ are disjoint, the expression in (1) is unique for each predicate $$P\in\mathcal{P}$$.