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Zk y ∨ z1 z2 . . zk y ∨ ψ1 ∨ ψ2 ∨ . . 2. Procedure Expand. 6 Orthogonal DNFs and number of true points Aclassical problem of Boolean theory is to derive an orthogonal disjunctive normal form of an arbitrary Boolean function. 13) j ∈Bk where Ak ∩ Bk = ∅ for all k = 1, 2, . . , m. 13. 13) is said to be orthogonal, or to be a sum of disjoint products, if (Ak ∩ B ) ∪ (A ∩ Bk ) = ∅ for all k, ∈ {1, 2, . . , m}, k = . 13 simply states that every two terms of an orthogonal DNF must be “conflicting” in at least one variable; that is, there must be a variable that appears complemented in one of the terms and uncomplemented in the other term.
Xn ) ∈ Bn . If CAB (X) = 1, then xi = 1 for all i ∈ A and xj = 0 for all j ∈ B, so that xi = 1 for all i ∈ F and xj = 0 for all j ∈ G. Hence, CF G (X) = 1 and we conclude that CAB implies CF G . To prove the converse statement, assume for instance that F is not contained in A. Set xi = 1 for all i ∈ A, xj = 0 for all j ∈ A and X = (x1 , x2 , . . , xn ). Then, CAB (X) = 1 but CF G (X) = 0 (since xk = 0 for some k ∈ F \ A), so that CAB does not imply CF G . 16. Let f be a Boolean function and C be an elementary conjunction.
For example, if (x1 , x2 , x3 ) = (0, 0, 0), then one successively finds that the state of each NOT-gate is 1 (= 1 − 0); the state of the AND-gate is 1 (= min(1, 1)); and the state of the output gate is 1 (= max(1, 0)). More generally, the gates of a combinational circuit may be “primitive” Boolean functions forming another class from the {AND,OR,NOT} collection used in our small example. In all cases, the gates may be viewed as atomic units of hardware, providing the building blocks for the construction of larger circuits.