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# Class Note for ENGIN 112 at UMass(28)

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Date Created: 02/06/15

ENGIN 112 Intro to Electrical and Computer Engineering Lecture 6 More Boolean Algebra Bi 2 ET AB ENGN112 L6 More Boolean Algebra September 15 2003 Overview Expressing Boolean functions Relationshi s between algebraic equations symbols and truth ta les Simplification of Boolean expressions Minterms and Maxterms ANDOR representations Product of sums Sum of products ENGN112 L6 More Boolean Algebra September 15 2003 Boolean Functions Boolean algebra deals with binary variables and logic operations Function results in binary 0 or1 x 2 F o oo o o 01 o X 010 o y 3 o 1 1 o I gt FXltz 100 1 z D 2 y y 101 o 110 1 111 1 Fxltyz Euclimz L5 Nhre nnlun upquot swimmmnna Boolean Functions Boolean algebra deals with binary variables and logic operations Function results in binary 0 or1 X ZX 00000 00100 x xy o1ooo y I 01101 Gxyyz 10000 2 gt 39z 1o1oo V 11010 11111 We will learn howto transition between equation symbols and truth table Euclimz L5 Nhre nnlun upquot swimmmnna Representation Conversion 0 ENGN112 L6 More Boolean Algebra Need to transition between boolean expression truth table and circuit symbols Converting between truth table and expression is easy Converting between expression and circuit is easy More difficult to convert to truth table September 15 2003 Truth Table to Expression 0 Converting a truth table to an expression Each row with output of 1 becomes a product term Sum product terms together x i Z G Any Boolean Expression can be 0 0 0 0 represented in sum of products form 0 0 1 0 0 1 0 0 0 1 1 1 1 0 0 0 1 0 1 0 1 1 0 1 1 1 1 1 1 I xyz xyz x yz ENGN112 L6 More Boolean Algebra September 15 2003 Equivalent Representations of Circuits All three formats are equivalent Number of 1 s in truth table outBut column equals AND terms for Sumof Products SO F ON Dag oooo oo oo A xoo xooolc G xyz xyz x yz ENGN112 L6 More Boolean Algebra September 15 2003 Reducing Boolean Expressions sthis the smallest possible implementation of thIs expression No 3 xyz Xyz X yz Use Boolean Algebra rules to reduce complexity while preserving functionality Step 1 Use Theorum 1 a a a o So xyz xyz x yz xyz xyz xyz x yz Step 2 Use distributive rule ab c ab ac So xyz xyz xyz x yz xyz z yzx x Step 3 Use Postulate 3 a a 1 So xyz z yzx x xy1 yz1 Step 4 Use Postulate 2 a 1 a Soxy1 yz1 xy yz xyz xyz x yz ENGN112 L6 More Boolean Algebra September 15 2003 Reduced Hardware Implementation 0 Reduced equation requires less hardware 0 Same function implemented F ON ELD G 3T A xoo xooolc G xyz xyz x yz xy yz ENGN112 L6 More Boolean Algebra September 15 2003 Minterms and Maxterms Each variable in a Boolean expression is a literal Boolean variables can appear in normal x or complement form x Each AND combination of terms is a minterm Each OR combination of terms is a maxterm For example For example Minterms Maxterms x y z Minterm x y z Maxterm O O O xyz m0 0 O O XyZ M0 0 O 1 xyz m1 0 O 1 Xyz M1 1 o o xy z m4 391quot o o x yz M4 1 1 1 xyz m7 1 1 1 x y Z M7 ENGN112 L6 More Boolean Algebra September 15 2003 Representing Functions with Minterms O Minterm number same as row position in truth table starting from top from 0 O Shorthand way to represent functions x 12 G 0 00 0 Gxzxz x z o 01 o y y y 010 o i 011 1 1 0 0 0 Gm7m6m32367 101 0 1 10 1 1 11 1 ENGN112 L6 More Boolean Algebra September 15 2003 Complementing Functions 0 Minterm number same as row position in truth table starting from top from 0 O xyzG 0000 0010 0100 0111 1000 1010 1101 1111 ENGN112 L6 More Boolean Algebra Shorthand way to represent functions G 1 G xyz xyz x yz G xyz xyz x yz Can we find a simpler representation OOAAOAA September 15 2003 Complementing Functions Step 1 assign temporary names bCgtZ Gabc az G G abc Step 2 Use DeMorgans Law az a z Step 3 Resubstitute bc for z a z a bc Step 4 Use DeMorgans Law G a b C 3 b C a b C G a b c a b c Step 5 Associative rule a b c a b c ENGN112 L6 More Boolean Algebra September 15 2003 Complementation Example Find complement of F x z yz F x z yz DeMorgan s F x z yz DeMorgan s F xZyZ Reduction gt eliminate double negation on x F xz y z This format is called product of sums ENGN112 L6 More Boolean Algebra September 15 2003 Conversion Between Canonical Forms Easy to convert between minterm and maxterm representations For maxterm representation select rows with 0 s y 0 2 OG G xyz xyz x yz 0 0 1 0 lt l 0 1 0 0 lt 0 1 1 1 Gm7m6m32367 1 0 0 0 4 l 1 0 1 0 1 1 0 1 G M0M1M2M4M5 l39l01245 1 1 1 1 i G XyZXyZ XY ZX yZX yZ ENGN112 L6 More Boolean Algebra September 15 2003 Representation of Circuits 0 All lo ic expressions can be represented in 2 level orma O Circuits can be reduced to minimal 2level representation 0 Sum of products representation most common in industry 21 Sum of Products b Product of Sums Fig 2 3 Two level implementation ENGN112 L6 More Boolean Algebra September 15 2003 Summary Truth table circuit and boolean expression formats are equwalent Easy to translate truth table to SOP and POS representation Booean algebra rules can be used to reduce circuit Size while maIntaInIng function girogic functions can be made from AND OR and Easiest way to understand Do examples Next time More logic gates ENGN112 L6 More Boolean Algebra September 15 2003

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