By Brandt A

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**Sample text**

The natural numbers form a well-ordered subset of the real numbers (N). Therefore, the real numbers are well-ordered. (f) If it is cold tomorrow (C), then I’ll wear my heavy coat (I) if the sleeve is mended (M). It will be cold tomorrow and the sleeve will not be mended. Therefore, I’ll not wear my heavy coat. (g) If the lottery is fixed (L) or the Colts leave town again (C), then the tourist trade will decline (D) and the town will suffer (S). If the tourist trade decreases, then the police force will be more content (P).

Non-validity was a notion that did not hold in formula variable form. In what follows, we again assume that we are working in formula variables. Of course, they also hold for atoms substituted for these variables. We use the notation for formula variables. Let A be a formula written in the following manner. The formula variables A1 , . . , An and only these formula variables are used to construct A with the L′ propositional connectives ¬, → . Thus A is written in formula variables. Since L′ ⊂ L, the truth-table concept can be applied to L′ .

The basic theorem used for all logic circuits is below. 5 If A ≡ B, then any logic circuit that corresponds to A can be substituted for any logical circuit that corresponds to B. Proof. Left to you. 4 Below are diagrams for two logic circuits. For the first circuit, note that if no circuit flows into lines A and B, then there is a current flowing out the C line. In current flows in the A and B line from left to right, then again current flows out the C line. But if current flows in the A and not in the B, or in the B and not in the A, then no current flows out the C line.