Abriss einer Theorie der algebraischen Funktionen einer by Hermann Stahl

By Hermann Stahl

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Verify: A -. B 7B DEMONSTRATION: (1) A -. B premise (2) 1 B > A equivalent to (I) by the law of the contrapositive (3) 1 B premise (4) 1 A Detachment; 3,2. -+ Ewapk2. Verify: :. 6 premise Detachment; 8,7 52 ANALYSIS OF INFERENCES A chain of inferences like the one in the preceding example is called a deduction of the conclusion from the premises. Such deductions resemble mathematical proofs (for example, the proofs of Greek geometry). This is not to say, however, that all proofs in geometry are such simple deductions.

32 ALGEBRA OF LOGIC De Morgan's laws would then appear as + B)' = A'B' (AB)' = A' + B' (A Of course, for uncomplicated expressions you may prefer not to use the new arithmetical notation at all. It is entirely optional. A good way to reduce to a conjunctive normal form is to perform the following four steps: 1. and -. 2. Use de Morgan's laws repeatedly until the only negated terms are variable symbols. 3. Switch t o arithmetical notation. 4. Multiply out (as in ordinary algebra). Naturally, if you get a chance to simplify things along the way by using the absorption, idempotent, domination, or identity laws, so much the better.

The following logic forms are conjunctive forms: A (satisfies condition i) 1 B (satisfies condition ii) TCvBvA (satisfies condition iii) ( 7 C v B v A) A A A 1 B (satisfies condition iv) ( iA v B) A (C v A) A (B v C) (satisfies condition iv) The following logic forms are not conjunctive normal forms: A+B ( A A B)v i C (AvB)rr(iB4A) 7 (A * B) To help fix the idea in your mind, think of a conjunctive normal \ form as a logic form which looks like where the heavy-type letters denote logic forms no more complicated than single variable symbols or negations of variable symbols.

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