Design of Logic Systems by D. Protheroe;Douglas Lewin;D. Lewin

By D. Protheroe;Douglas Lewin;D. Lewin

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When implemented in terms of switching circuits (Fig. 3), the two forms of the function T given above have identical switching characteristics. Note, however, that the actual logic gates required to implement these functions are different: the sum-of-products requires two AND gates and one OR, while the product-of-sums needs two OR gates and one AND gate. In these two alternative forms of switching function, all the binary variables or their complements (literals) appear once, and only once, in each term or factor.

The sets AI = {I, 2}, A2 = {3, 4}, A3 = {5, 6, 7}, A4 = {8, 9} are disjoint and hence form a partition P = (1,2) (3, 4) (5,6, 7) (8,9). The basic operations obey the following algebraic laws: 1. Commutative law 2. Associative law 3. Distributive law 4. Absorption law 5. Idempotent law 6. Complement law 7. Universal and null law A n B = B n A; A U B = B U A A n (B n C) = (A n B) n C A U (B U C) = (A U B) U C A n (B U C) = (A n B) U (A n C) A U (B n C) = (A U B) n (A U C) A n (A U B) = A U (A n B) = A AnA=AuA=A A U A = I; A n A = <1>; (A) = A A n B = A U 8; A U B = A n 8 n A = <1>, U A = A, I nA I UA (De Morgan's Theorem) = A = I where A, Band C are subsets of some universal set I, and is the null set.

B_O_O_L_EA_N __AL __G_EB_RA __________________~I L -_ __ __ _ _ __ __ __ __ _ Universal set I o (a) A c: I (b) A uB = B uA (e) A n B = B n A r------, I 66 (e) A n B (I) A n (B n C) = (A n B) n C = A and B are said to be disjoint J A n (B u C) (A n B) u (A n C) (g) A n (B u C) = (A n B) u (A n C) J AnB (h) AU B = Fig. 1 An S (i) I = (A n S)U(A n B)u(A n S)U(A n B) Venn diagram proofs. 2. e. a + b = b + a and a. b = b . a. 3. e. a(b + e) = ab + ae and a + be = (a + b) . (a + e). 4. Identity elements must exist such that 0 + a = a and 1 .

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