Lecture 3 - Boolean Algebra

# Operations 20 nand 0 1 nor 0 1 xor 0 1 0 1 1 0 1 0 0

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Unformatted text preview: x, x · x = x (P2) Identity: x + 0 = x, x · 1 = x (P3) Domination: x + 1 = 1, x · 0 = 0 (P4) Complement: x + x = 1, x · x = 0 ¯ ¯ (P5) Involution: x = x. ¯ Proofs by perfect induction January 24, 2013 Lecture 3 AWB More properties (P6) Commutativity: x + y = y + x, x · y = y · x (P7a) Associativity: x + (y + z ) = (x + y ) + z , (P7b) Associativity: x · (y · z ) = (x · y ) · z (P8) Covering x + x · y = x, x · (x + y ) = x (P9a) Distributivity: x · (y + z ) = (x · y ) + (x · z ), (P9b) Distributivity: x + (y · z ) = (x + y ) · (x + z ) (10a) Combining: x·y+x·y =x ¯ (10b) Combining: 22 (x + y ) · (x + y ) = x ¯ January 24, 2013 Lecture 3 AWB Distributivity - by perfect induction 23 Prove property (P9b): x + (y · z ) = (x + y ) · (x + z ) x y z L = x + (y · z ) R = (x + y ) · (x + z ) 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 1 1 1 1 1 0 0 1 1 1 0 1 1 1 1 1 0 1 1 1 1 1 1 1 January 24, 2013 Lecture 3 L=R AWB Distributivity - from properties 24 (P9b) also follows from (P1)-(P9a) R = (x + y ) · (x + z ) = (x + y ) · x + (x + y ) · z (P 9 a ) = x · (x + y ) + z · (x + y ) (P 6)) = x · x + x · y+z · x+ z · y (P 9 a ) = (P 9 a ) x +x · y + x · z + y · z = x + y · z (P 1) January 24, 2013 (P 6) (P 6) Lecture 3 (P 6) (P 6)+(P 8) AWB Completeness of NAND and NOR NOT A A AND B A OR B A NAND A NOT( A NAND B ) (NOT A) NAND (NOT B) A NOR A (NOT A) NOR (NOT B) 25 NOT (A NOR B) We can built circuits from NAND or NOR gates only! January 24, 2013 Lecture 3 AWB De Morgan laws ¯¯ A+B =A·B A B 26 ¯¯ A·B =A+B A Y B A Y A Y B B A B January 24, 2013 Y A Y B Lecture 3 Y AWB De Morgan laws 27 In general: ¯¯ ¯ • A1 + A2 + · · · + An = A1 · A2 · · · An . . . ¯ ¯ ¯ A1 · A2 · · · An = A1 + A2 + · · · An . . . January 24, 2013 . . . Lecture 3 AWB Bubble pushing 28 ’Bubble’ represents negation. When ’bubble’ on the output is pushed back to the inputs • OR chnages to AND, AND changes to OR • bubbles appear on inputs • two bubbles in sequence same as no bubble January 24, 2013 Lecture 3 AWB Bubble pushing A·B+C ·D 29 (A · B ) + (C · D ) A·B·C ·D A A A B B B Y Y Y C C C D D D January 24, 2013 Lecture 3 AWB Bubble pushing 30 A A B B C C D E F January 24, 2013 D Y Y E F Lecture 3 AWB Systematic design 31 Terminology: • Complement: negated variable • Literal: variable or its complement • Product term: logical AND of literals • Sum term: logical OR of literals • Minterm: product term where all input variables appear exactly once • Maxterm: sum term where all input variables appear exactly once January 24, 2013 Lecture 3 AWB Systematic design - minterms 32 minterm m0 = a¯c ¯b ¯ a b c s = F (a, b, c) 0 0 0 0 m1 = a¯ ¯bc 0 0 1 1 m2 = abc ¯¯ 0 1 0 1 Sum minterms m3 = abc ¯ m4 = a¯c b¯ 0 1 1 0 for which 1 0 0 1 F (a, b, c) = 1 m5 = a¯ bc 1 0 1 0 m6 = abc ¯ 1 1 0 0 m7 = abc 1 1 1 1 Canonical SOP form: F (a, b, c) = m1 + m2 + m4 + m7 January 24, 2013 Lecture 3 AWB Two-level logic F (a, b, c) = 33 m1 + m2 + m4 + m7 = { 1 ,2 ,4 ,7 } a¯ + abc + a¯c + abc ¯bc ¯ ¯ b¯ = Two-level logic (AND array followed by OR gate): a b c m1 m2 s m4 m7 January 24, 2013 Lecture 3 AWB FA - bit of carry minterm m0 = a¯c ¯b¯ a b cin cout = F (a, b, cout ) 0 0 0 0 m1 = a¯ ¯bc 0 0 1 0 m2 = abc ¯¯ 0 1 0 0 m3 = abc ¯ m4 = a¯c b¯ 0 1 1 1 1 0 0 0 m5 = a¯ bc 1 0 1 1 m6 = abc ¯ 1 1 0 1 m7 = abc 1 1 1 34 1 F (a, b, cin ) = m3 + m5 + m6 + m7 January 24, 2013 Lecture 3 AWB Systematic design - FA F (a, b, cin ) = 35 m3 + m5 + m6 + m7 = { 3 ,5 ,6 ,7 } abc + a¯ + abc + abc ¯ bc ¯ = a b cin m3 m5 cout m6 m7 January 24, 2013 Lecture 3 AWB Systematic design - simpliﬁcation 36 F (a, b, c) = abc + a¯ + abc + abc ¯ bc ¯ bc ¯ = abc + abc + a¯ + abc + abc ¯ = (¯ + a)bc + a¯ + abc + abc + abc a bc ¯ = bc + (¯ + b)ac + abc + abc b ¯ = bc + ac + ab(¯ + c) c = bc + ac + ab Is there any other way? January 24, 2013 Lecture 3 AWB Next .... 37 More Boolean algebra Karnaugh maps Read Sections 2.1-2.5 January 24, 2013 Lecture 3 AWB...
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## This note was uploaded on 09/26/2013 for the course ECE 2300 taught by Professor Long during the Fall '08 term at Cornell University (Engineering School).

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