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not (A or B) = (not A) and (not B) not (A and B) = (not A) or (not B) where "A or B" is an "inclusive or" meaning at least one of A or B rather than an "exclusive or" that means exactly one of A or B. De Morgan's law with set subtraction operation. Another form of De Morgan's law is the following as seen below.
For example, the formula "a AND NOT b" is satisfiable because one can find the values a = TRUE and b = FALSE, which make (a AND NOT b) = TRUE. In contrast, "a AND NOT a" is unsatisfiable. SAT is the first problem that was proven to be NP-complete—this is the Cook–Levin theorem.
In the truth table below, d1 is the formula: ( (IF c THEN b) AND (IF NOT-c THEN a) ). Its fully reduced form d2 is the formula: ( (c AND b) OR (NOT-c AND a). The two formulas are equivalent as shown by the columns "=d1" and "=d2". Electrical engineers call the fully reduced formula the AND-OR-SELECT operator.
The exclusive or does not distribute over any binary function (not even itself), but logical conjunction distributes over exclusive or. C ∧ ( A ⊕ B ) = ( C ∧ A ) ⊕ ( C ∧ B ) {\displaystyle C\land (A\oplus B)=(C\land A)\oplus (C\land B)} (Conjunction and exclusive or form the multiplication and addition operations of a field GF(2 ...
Both A and B conjunct A and B are conjoined Disjunction Either A or B, or both disjunct A and B are disjoined Negation It is not the case that A negatum/negand A is negated Conditional If A, then B antecedent, consequent B is implied by A Biconditional A if, and only if, B equivalents A and B are equivalent
Definition 2: If is a propositional connective, and A, B, C, … is a sequence of m, possibly but not necessarily atomic, possibly but not necessarily distinct, formulas, then the result of applying to A, B, C, … is a formula.
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In mathematical logic (a subtopic within the field of formal logic), two formulae are equisatisfiable if the first formula is satisfiable whenever the second is and vice versa; in other words, either both formulae are satisfiable or both are not. [1] Equisatisfiable formulae may disagree, however, for a particular choice of variables.