From formulas to truth regions

A truth table gives a formula a true or false value at each complete assignment. Today we collect all the true rows into a set. Keep our four states fixed. A has a dot only, B has both features, C has a stripe only, and D has neither. X means dot, and Y means stripe. X is true at A and B. Its truth region is therefore the set containing A and B. The paper's symbol on screen names this truth-region operation. Keep three things separate: X is a formula; true is its value at A; and the set containing A and B is its truth region. We are describing every possible state, not announcing which hidden card we have. X is true at A and B. Y is true at B and C. For X and Y, we need both requirements in the same state. Which state belongs to both input regions? Only B. That shared part is the intersection. So the truth region of X and Y is the singleton containing B. The rule works for any two formulas: a state satisfies their AND exactly when it belongs to both truth regions. That is why AND becomes intersection. For X or Y, at least one requirement is enough. A qualifies through X. B qualifies through both. C qualifies through Y. D qualifies through neither. The resulting region contains A, B and C. This is the union of the two input regions. B is included once, even though both inputs include it. Inclusive OR keeps the overlap. We will soon compare it with exclusive-or, which removes that overlap. Not X is true exactly where X is false. In this universe, that is C and D. Its truth region is the complement of the X region inside our fixed carrier. Likewise, not Y is true at A and D. C has a stripe, so C does not satisfy not Y. Intersect X with not Y. A is in both regions; B fails not Y; C and D fail X. So X and not Y describes A alone. Complement always means outside the chosen set but still inside the stated universe. Describing the no-dot region does not, by itself, give an observer a no-dot certificate. Exclusive-or, or XOR, is true when the two inputs differ. The double vertical arrow is our written symbol for it. At A, X is one and Y is zero: they differ, so XOR is one. At B, both are one: they agree, so XOR is zero. At C, zero and one differ, so XOR is one. At D, both are zero, so XOR is zero. We have now checked every row. The XOR truth region contains A and C. Compare it with OR, which also contains B. OR accepts one or both features; XOR accepts exactly one. The union splits into two disjoint pieces: the overlap, B, and the exclusive remainder, A and C. Disjoint means the pieces share no state. For any two sets, the part in one but not both is called their symmetric difference. That is the set operation corresponding to XOR. Its triangle symbol here means symmetric difference, not a boundary. Two formulas are semantically equivalent when they have the same truth value at every valuation. Now compare X with X or Y. Their truth regions overlap. Does that make the formulas equivalent? If not, name a state where their values differ. No. At C, X is false while X or Y is true. Overlap is not equality. C is a counterexample to equivalence, and the symmetric difference here is the singleton containing C. Compare X or Y with this second expression: NOT, applied to the whole conjunction NOT X AND NOT Y. Work from the inside outward. First evaluate the two inner negations and their AND; only then reverse that AND result with the outer NOT. The expressions are written differently. Do they describe the same Boolean rule? At A, not X is false, so the inner AND is false and its negation is true. At B the inner AND is again false. At C, not Y is false, so the result is true again. At D, both inner negations are true. Their AND is true, so its negation is false. Both formulas therefore have outputs one, one, one, zero in our row order. Their truth regions are equal, so the formulas are semantically equivalent: they agree at every valuation in the fixed universe. Their written expressions are still different. Why is this check enough? Our table contains every assignment of X and Y. One differing row would disprove equivalence. Complete agreement over all four rows proves it for these two-input formulas. More generally, equal truth regions mean no valuation makes exactly one formula true. Their XOR region is empty. A nonempty XOR region supplies a counterexample. Try a new comparison. Is Y and not X equivalent to X exclusive-or Y? Find their truth regions and a counterexample if they differ. They differ. Y and not X is true only at C. XOR is true at A and C. A is a counterexample: XOR is true there, but Y and not X is false. We can now translate AND into intersection, OR into union, NOT into complement, and XOR into symmetric difference. Equal truth regions characterize semantic equivalence. Next, complete descriptions will show how individual states build every finite truth region. We will then develop distance and topology before choosing which truth regions count as available observations.