Partitions, minterms, and complete descriptions

Our cards use X for a dot and Y for a stripe. A has only a dot; B has both; C has only a stripe; D has neither. In each two-bit annotation, X comes first and Y second. One means true, and zero means false. A formula’s truth region is the set of cards where it is true. Imagine sorting all four state cards into boxes. Every card must enter a box, no card may enter two boxes, and no box is empty. Such a collection of boxes is called a partition of the carrier. The dot states A and B make one block. The no-dot states C and D make the other. Their union is the whole carrier, and their intersection is empty. This is one partition, with two blocks rather than four points. Now describe A completely: X is true and Y is false. The formula X and not Y selects A alone. A conjunction that specifies a truth value for every variable is called a complete minterm. B requires both variables true. C requires not X and Y. D requires not X and not Y. These descriptions are mutually exclusive: no state satisfies two of them. They are exhaustive: every state satisfies one. Together, they describe every complete assignment exactly once. Suppose our target is the set containing A and C. Start with the complete description of A. It contributes the singleton containing A, rather than the entire region where X is true. The complete description of C contributes the other singleton. A state belongs to the target when it satisfies either description. The word either here is inclusive OR; disjointness makes simultaneous satisfaction impossible anyway. The disjunction therefore defines exactly A and C. XOR means exactly one of the two inputs is true. This is our XOR region. We built it from complete descriptions instead of guessing a shorter formula and hoping its truth region matched. The power set means the collection of every subset of the carrier. Each of four points can independently be included or omitted, giving sixteen subsets. A point, a subset, and this collection of subsets are different kinds of object. To describe any target subset in this finite-variable setting, take the complete minterm of each point it contains and join those minterms by OR. A point satisfies that formula exactly when it is one of the chosen points. That explains why the construction works for every subset of our finite valuation carrier. For the empty target there are no chosen minterms, so the empty disjunction means false. An empty conjunction imposes no requirements and means true. These conventions keep the same construction valid at the extreme cases. Two formulas are semantically equivalent when they agree at every valuation. For this two-variable carrier, checking all four rows is exhaustive. Checking A alone proves only agreement at A, not equivalence over the carrier. The familiar union, intersection and complement operations on the power set form a finite Boolean algebra. Here that name packages operations we already understand; it does not add a new rule of inference. Complete minterms are the atomic pieces of classical truth sets. A definable singleton, written as the set containing v, is one valuation named by a formula. This does not automatically make singletons a basis for an observation topology. If the available observations cannot certify that singleton, logical definability alone does not make it open. Reconstruct the set containing B and D from complete minterms. State the two conjunctions and explain why the other two valuations are excluded. Pause before revealing the formula. B contributes X and Y. D contributes not X and not Y. Their disjunction is true at B and D. At A and C the variables disagree, so neither complete conjunction is true. We can now move in both directions: from a formula to its truth region, and from any finite truth region to a formula. Later, observations will select which of those definable events can be certified.