By Gilbert Demengel

ISBN-10: 2040012648

ISBN-13: 9782040012649

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Extra resources for Aide Mémoire Analyse Mathématique

Sample text

Xn ) or ψ(x1 , x2 , . . , xn ) to denote Boolean expressions in the variables x1 , x2 , . . , xn . 2. Here are some examples of Boolean expressions: φ1 (x) = x, φ2 (x) = x, ψ1 (x, y, z) = (((x ∨ y)(y ∨ z)) ∨ ((xy)z)), ψ2 (x1 , x2 , x3 , x4 ) = ((x1 x2 ) ∨ (x 3 x 4 )). Now, since disjunction, conjunction, and complementation can be interpreted as Boolean functions, every Boolean expression φ(x1 , x2 , . . , xn ) can also be viewed as generating a Boolean function defined by composition. 5. The Boolean function fφ represented (or expressed) by a Boolean expression φ(x1 , x2 , .

As a matter of fact, f has no prime implicants other than xy and xz. 7 Implicants and prime implicants 27 Prime implicants play a crucial role in constructing DNF expressions of Boolean functions. 4). 13. Every Boolean function can be represented by the disjunction of all its prime implicants. Proof. Let f be a Boolean function on B n , and let P1 , P2 , . . , Pm be its prime implicants (notice that m is finite because the number of elementary conjunctions on n variables is finite). Consider any DNF representation of f , say φ = rk=1 Ck .

Xn ) = f (x1 , . . , xk−1 , 1, xk+1 , . . 8 Restrictions of functions, essential variables 29 and similarly for f|xk =0 (x1 , x2 , . . , xn ). This slight abuse of definitions is innocuous and we use it whenever it proves convenient. Also, we use shorthand like f|x1 =0,x2 =1,x3 =0 instead of the more cumbersome notation f|x1 =0 |x =1 . 2 |x3 =0 The link between representations of a function and representations of its restrictions is straightforward. 15. Let f be a Boolean function on Bn , let ψ be a representation of f , and let k ∈ {1, 2, .