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五、狄、摩根定理的扩充——互补定理 (一)狄、摩根定理的证明 狄、摩根定理有两种表达形式,即: A+B+C=A·B·C ……(5.1) ABC=A+B+C ……(5.2) 其实狄、摩根定理是显而易见的互补定理的特例。根据补码关系可知,若M_1=M_2,则必有M_1+M_2≡1。据此我们可以改写(5.1)式为: A+B+C+A·B·C≡1 ……(5.3) (5.3)式的逻辑含义十分明显,由 A、B、C三变数构成“或”逻辑(A+B+C),显然它包括除了A·B·C“和”以外的全部可能组合,因而(A+B+C)与A·B·C互补,于是(5.3)式成立。 同理亦可证明(5.2)式的关系。
V. EDITORS 'EXPANSION - COMPLEMENTARY THEOREM (1) Di and Morgan's Proof Di and Morgan's theorem have two forms of expression: A + B + C = A · B · C (5.1) ABC = A + B + C ...... (5.2) In fact Di, Morgan's theorem is obvious case of complementary theorems. According to the complement relationship, we can see that if M_1 = M_2, M_1 + M_2≡1 must exist. Therefore, we can rewrite the equation (5.1) as: A + B + C + A · B · C≡1 (5.3) The logical meaning of (5.3) is obvious and consists of three variables A, B, (A + B + C), it is clear that it includes all possible combinations except for A · B · C “and” and therefore (A + B + C) is complementary to A · B · C . Similarly, the relationship between (5.2) can also be proved.