Computer Science and Multiple-Valued Logic. Theory and by David C. Rine

By David C. Rine

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Extra resources for Computer Science and Multiple-Valued Logic. Theory and Applications

Example text

This modification consists of minor circuit alterations and a different biasing scheme in which R2 < R0 and R2

The remaining parts of the theorem hold because if i ^ k ^ j , then ek is in the sublattice generated by B' U { / 0 , . . ,/r-i}. 6. Let A be a bounded distributive lattice with center B, and x, y, z G A. (i) If x-*z and y -* z ejcwf, fnen (JC Vy) -► z = (x -> z)(y -> z). (ii) If z-*x and z-+y exist, then z -> xy = (z -> JC)(Z -» y). (iii) 7/ JC -> y exists, b GB and c ELB, then bx-*(c vy) = f v c V(JC -> y). (iv) 7/ JC => z and y => z ejcisf, f/ien (JC v y) => 2 = (JC => z)(y ^ z). (v) If z => JC and z ^ y ejcisf, fnen z => jcy = (z => x)(z => y).

N - 1 , by Theorem 9. Secondly, suppose that w ss Dj(Xi) for some fixed / = 1, 2 , . . , n - 1 and all i G J. Let w' = C„-i(w) ve,-_i. Then it follows from Theorem 12, and from Theorem 3 and Theorem 4, that C-i(w') = Cn_,(w), and also that C(w') = 0 for / = /, j + 1 , . . , n - 2. Hence A ( H O = V G(w') = C„_,(w). 32 Ch. 2 GEORGE EPSTEIN It is clear from Axiom 3 that, for each i e l , JC,- ^^-iv£>,(*,). Since w ^Dj(Xi), Cn-i(w)^Cn-i(Dj(Xi)) = D}(Xi) by Theorem 9 and Theorem 3. Hence JCi =SS ei-lvCn-i(w) = w' for all j GJ.

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