ece15_4_2011_6

# ece15_4_2011_6 - Today Boolean functions Constant a symbol...

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1 ECE 15A Fundamentals of Logic Design Lecture 4 Malgorzata Marek-Sadowska Electrical and Computer Engineering Department UCSB 2 Today : Boolean functions Constant – a symbol which represents a specified element of Boolean algebra Variable: a symbol a,b,c,…etc. representing unspecified elements Literal - a variable with specified polarity: a,a’,b, b’,…etc. Boolean function: any expression which represents the combination of finite set of symbols, each representing a constant or variable, by the operations of (+), ( ), or (‘). Examples: F1(a,b,c,d,x)=(a+b)c’ + (a+b’x)d F2(a,b,c,d)= ab’c+a’d 3 Disjunctive normal form Definition. A Boolean function is in disjunctive normal form in n variables x1,x2,…,xn, for n>0, if The function is a sum of terms where No two terms are identical 0 and 1 are in disjunctive normal form in n variables for any n>0. Example: f(a,b,c,d) = ab’cd+a’bc’d+abcd’ The terms of DNF are called minterms ) ( i i i x f = i i i i x x x f i ' ) ( 4 Theorem 1a In a Boolean algebra, every function which contains no constants can be represented in a disjunctive normal form. Proof. Suppose that f(x1,x2,…,xn) is not in dnf. If it contains expression (g+h)’ or (gh)’ for some functions g,h, then they can be written as g’h’ and g’+h’. This process may be continued until each (’) applies to a single variable xi. Next, we can apply distributive law of ( ) over (+) and express f as a polynomial. Each term tj with a missing variable xi can be expressed as tj(xi+xi’). Using the laws of Boolean algebra, we can eliminate duplicate terms. 5 Example f(a,b,c)=(ab’+ac)’+b’ =(ab’)’(ac)’+b’=(a’+b)(a’+c’)+b’= a’+a’b+a’c’+bc’+b’= a’(b+b’)(c+c’)+ a’b(c+c’) + a’(b+b’)c’ +(a+a’)bc’+(a+ a’)b’(c+c’)= a’bc+a’bc’+a’b’c+a’b’c’+abc’+ a’bc’ +ab’c+ab’c’+ a’b’c +a’b’c’ = a’bc+a’bc’+a’b’c+a’b’c’+abc’+ab’c+ab’c’ 6 Disjunctive normal form Unique representation of a function in a given number variables Example: f(a,b) = a’b f(a,b,c) = a’b = a’b(c’+c) = a’bc+a’bc’ We will assume that disjunctive normal form refers to a dnf which contains the fewest possible number of variables.

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