CK Cheng Tuesday 10/2/02 CS 140 Lecture 2. Part I. Combinational Logic I) Specification –a. Language –b. Truth Table –c. Boolean Algebra –d. Incompletely.

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Presentation transcript:

CK Cheng Tuesday 10/2/02 CS 140 Lecture 2

Part I. Combinational Logic I) Specification –a. Language –b. Truth Table –c. Boolean Algebra –d. Incompletely Specified Function

Binary Addition Carry Sum Carryout Sums Carry bits

Half Adder a b carry sum Truth Table a b Sum Carry

Switching function Sum (a,b) = a’b + ab’ Ex: Sum (0,0) = 0’0 + 0*0’ = = 0 Sum (0,1) = 0’1 + 0*1’ = = 1 Sum (1,1) = 1’1 + 1*1’ = = 0 Carry (a, b) = a*b a b sum a b carry

Full Adder Id a b c in carry sum Truth Table a b Sum Carry C in

Minterm and maxterm Id a b c in carryout a+b+c a+b+c’ a+b’+c a’ b c a’+b+c a b’c a b c’ a b c minterm maxterm

f 1 (a,b,c) = a’bc + ab’c + abc’ + abc a’bc = 1 iff (a,b,c,) = (0,1,1) ab’c = 1 iff (a,b,c,) = (1,0,1) abc’ = 1 iff (a,b,c,) = (1,1,0) abc = 1 iff (a,b,c,) = (1,1,1) f 1 (a,b,c) = 1 iff (a,b,c) = (0,1,1) or (1,0,1) or (1,1,0) or (1,1,1) Minterms Ex: f 1 (1,0,1) = 1’ ’ ’ = 1 f 1 (1,0,0) = 1’ ’ ’ = 0

f 2 (a,b,c) = (a+b+c)(a+b+c’)(a+b’+c)(a’+b’+c) a + b + c = 0 iff (a,b,c,) = (0,0,0) a + b + c’ = 0 iff (a,b,c,) = (0,0,1) a + b’ + c = 0 iff (a,b,c,) = (0,1,0) a’ + b + c = 0 iff (a,b,c,) = (1,0,0) f 2 (a,b,c) = 0 iff (a,b,c) = (0,0,0) or (0,0,1) or (0,1,0) or (1,0,0) Maxterms Ex: f 2 (1,0,1) = (1+0+1)(1+0+1’)(1+0’+1)(1’+0+1) = 1 f 2 (0,1,0) = (0+1+0)(0+1+0’)(0+1’+0)(0’+1+0) = 0

f 1 (a,b,c) = a’bc + ab’c + abc’ + abc f 2 (a,b,c) = (a+b+c)(a+b+c’)(a+b’+c)(a’+b’+c) f 1 (a, b, c) = m3 + m5 + m6 + m7 =  m(3,5,6,7) f 2 (a, b, c) = M0M1M2M4 =  M(0, 1, 2, 4) Does f 1 = f 2 ? Yes! Prove using Boolean algebra.

Id a b c in carry minterm 4 = ab’c’ Minterm 4 = ab’c’ Only one row has a 1.

Id a b c in carry maxterm 4 = a+b+c Maxterm 4 = a’+b+c Only one row has a 0.

Incompletely Specified Function Id a b c g(a,b,c) g 1 (a,b,c) canonical form w/ minterms g 2 (a,b,c) canonical form w/ maxterms Does g 1 (a,b,c) = g 2 (a,b,c)? No! Because there is a group of Don’t Care set. (g 1 only covers the onset, g 2 only covers the offset).

Id a b f (a, b) )The input does not happen. 2)The input happens, but the output is ignored. Examples: -Decimal number 0… 9 uses 4 bits. (1,1,1,1) does not happen. -Final carry out bit (output is ignored). Don’t care set is important because it allows us to minimize the function