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l4_15a_6 - Last time We have proved all fundamental laws...

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1 ECE 15A Fundamentals of Logic Design Lecture 4 Malgorzata Marek-Sadowska Electrical and Computer Engineering Department UCSB 2 Last time : We have proved all fundamental laws stated for sets. Theorem 1 : Every statement or algebraic identity deducible from P1- P4 remains valid if the operations (+) and ( ), and the identity elements 0 and 1 are interchanged throughout. Theorem 2: For every element a in Boolean algebra B: a+a =a and aa = a. Theorem 3 : For each element a in Boolean algebra B, a + 1 = 1 and a 0 = 0. Theorem 4 . For each pair of elements a, b in a Boolean algebra B, a + ab = a and a(a + b) = a. Theorem 5 . In every Boolean algebra B each of the binary operations (+) and ()is associative. That is, for every a, b and c in B: a + (b + c) = (a + b) + c and a(bc) = (ab)c 3 Last time: cont. Theorem 6. The element a’ associated with element a in a Boolean algebra is unique. Theorem 7. For every a in a Boolean algebra B, (a’)’=a. Theorem 8 . In any Boolean algebra, 0’ = 1 and 1’ = 0. Theorem 9. For every a and b in Boolean algebra B, (ab)’ = a’ + b’ and (a+b)’ = a’b’ [De Morgan’s laws] 4 Example: 2-valued Boolean algebra Boolean algebra on B={0,1} and operators {+, ,’} From theorem 2 [a+a=a, aa=a]: 0+0 = 0 1+1 = 1 0 0 = 0 1 1 = 1 From Theorem 3 [a+1=1, a0=0]: 0+1=1 0 0 = 0 1+1 = 1 1 0 = 0 From Theorem 8: 0’ = 1 1’ = 0 + 0 1 0 0 1 1 1 1 0 0 0 0 1 1 0 1 5 Today: Boolean functions B={0,1} B is a set of all tuples (b1, b2,… bk), such that bi is either 0 or 1 for all 1<=i<=k. B may be viewed as the set of all k-bit strings. Example: k = 3. B = {(0,0,0), (0,0,1), (0,1,0), (0,1,1),(1,0,0),(1,0,1),(1,1,0),(1,1,1)}. B has a size of 2 . k k 3 k k 6 Boolean function Boolean function f: B -> B , k>0, m>0. k-bit input is mapped to an m-bit output k m B B k m mapping Domain Co-domain An element in a domain is assigned one element in the co-domain. m=1: single-output function.
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2 7 Boolean functions – algebraic form 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 of single-output functions: ¾ F1(a,b,c,d,x)=(a+b)c’ + (a+b’x)d ¾ F2(a,b,c,d)= ab’c+a’d 8 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 ) ( 1 i n i i x f = = i i i i x x x f i ' ) ( 9 Example f(a,b,c,d) = ab’cd+a’bc’d+abcd’ f(0,0,1,0)= 0 1 1 0 + 1 0 0 0 + 0 0 1 1 = 0+0+0=0 f(1,0,1,1)= 1 1 1 1 + 0 0 0 1 + 1 0 1 0 = 1+0+0=1
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