ece15_5_2011_6

# ece15_5_2011_6 - Canonical representations of Boolean...

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1 ECE 15A Fundamentals of Logic Design Lecture 5 Malgorzata Marek-Sadowska Electrical and Computer Engineering Department UCSB 2 Canonical representations of Boolean functions Canonical forms are unique representations of Boolean functions. dnf and cnf are canonical representations of Boolean functions To convert between dnf and cnf, we interchange the sum and product symbols and list those numbers which were absent in the original form. = ) , , ( z y x F (m3,m5,m6,m7) = ) , , ( z y x F (M0,M1,M2,M4) Example: 3 Boolean functions in standard forms Sum-of-products (SOP) Boolean expression that contains AND terms (called products or implicants) which are ORed together Example: F1(x,y,z,w)=xy+z’+x’w’w’ Product of sums (POS) Boolean expression that contains OR terms (called sum terms or implicants) which are ANDed together Example: F2=(x+z’+w)(x’+y+w’) These normal forms are not unique: the same functions can be expressed in many different ways Standard forms usually contain fewer terms and literals than the corresponding dnfs or cnfs. 4 XOR operation 0 1 0 1 1 1 0 A B = B A (commutative) (A B) C = A (B C) (associative) (AB) (AC) = A (B C) multiplication distributive over L = (A B)(A C) = A (BC)= R A B C A B A C L BC R 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 1 0 1 0 0 0 0 0 1 1 1 1 1 1 1 1 0 0 1 1 1 0 1 1 0 1 1 0 0 0 1 1 1 0 0 1 0 0 1 1 1 1 0 0 0 1 0 A C = B B C = A A B C = 0 If A B = C, then ? 5 Other logic operations NOR f(a,b) = (a+b)’ NAND f (a,b) = (ab)’ XNOR (Equivalence) f (a,b)= ab + a’b’ NOR 0 1 0 1 0 1 0 0 NAND 0 1 0 1 1 1 1 0 XNOR 0 1 0 1 0 1 0 1 6 Functionally complete operations A set of operations is functionally complete (universal) if and only if every Boolean function can be expressed entirely by means of operations from this set. We have shown that every function can be expressed as cnf and dnf, so (AND,OR,NOT) is functionally complete.

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## This note was uploaded on 02/26/2011 for the course ECE 15A taught by Professor M during the Winter '08 term at UCSB.

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ece15_5_2011_6 - Canonical representations of Boolean...

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