ENIN 2 Intro to Electrical and Computer Engineering Lecture 6 More Boolean Algebra ENIN2 L6: More Boolean Algebra September 5, 23 A B
Overview Epressing Boolean functions Relationships between algebraic equations, smbols, and truth tables Simplification of Boolean epressions Minterms and Materms AND-OR representations Product of sums Sum of products ENIN2 L6: More Boolean Algebra September 5, 23
ENIN2 L6: More Boolean Algebra September 5, 23 Boolean Functions Boolean algebra deals with binar variables and logic operations. Function results in binar or F F = (+ ) + F = (+ )
ENIN2 L6: More Boolean Algebra September 5, 23 Boolean Functions Boolean algebra deals with binar variables and logic operations. Function results in binar or = + We will learn how to transition between equation, smbols, and truth table.
Representation Conversion Need to transition between boolean epression, truth table, and circuit (smbols). Converting between truth table and epression is eas. Converting between epression and circuit is eas. More difficult to convert to truth table. Circuit Boolean Epression Truth Table ENIN2 L6: More Boolean Algebra September 5, 23
ENIN2 L6: More Boolean Algebra September 5, 23 Truth Table to Epression Converting a truth table to an epression Each row with output of becomes a product term Sum product terms together. + + An Boolean Epression can be represented in sum of products form!
ENIN2 L6: More Boolean Algebra September 5, 23 Equivalent Representations of Circuits All three formats are equivalent Number of s in truth table output column equals AND terms for Sum-of-Products (SOP) = + +
Reducing Boolean Epressions Is this the smallest possible implementation of this epression? No! = + + Use Boolean Algebra rules to reduce compleit while preserving functionalit. Step : Use Theorum (a + a = a) So + + = + + + Step 2: Use distributive rule a(b + c) = ab + ac So + + + = ( + ) + ( + ) Step 3: Use Postulate 3 (a + a = ) So ( + ) + ( + ) =. +. Step 4: Use Postulate 2 (a. = a) So. +. = + = + + ENIN2 L6: More Boolean Algebra September 5, 23
ENIN2 L6: More Boolean Algebra September 5, 23 Reduced Hardware Implementation Reduced equation requires less hardware! Same function implemented! = + + = +
Minterms and Materms Each variable in a Boolean epression is a literal Boolean variables can appear in normal () or complement form ( ) Each AND combination of terms is a minterm Each OR combination of terms is a materm For eample: Minterms Minterm m m m 4 m 7 For eample: Materms Materm ++ M ++ M ++ M 4 + + M 7 ENIN2 L6: More Boolean Algebra September 5, 23
ENIN2 L6: More Boolean Algebra September 5, 23 Representing Functions with Minterms Minterm number same as row position in truth table (starting from top from ) Shorthand wa to represent functions = + + = m 7 + m 6 + m 3 = Σ(3, 6, 7)
ENIN2 L6: More Boolean Algebra September 5, 23 Complementing Functions Minterm number same as row position in truth table (starting from top from ) Shorthand wa to represent functions = + + = ( + + ) = Can we find a simpler representation?
Complementing Functions Step : assign temporar names b + c -> (a+ ) = Step 2: Use DeMorgans Law (a + ) = a. = a + b + c = (a + b + c) Step 3: Resubstitute (b+c) for a. = a. (b + c) Step 4: Use DeMorgans Law a. (b + c) = a. (b. c ) Step 5: Associative rule a. (b. c ) = a. b. c = a + b + c = a. b. c = a b c ENIN2 L6: More Boolean Algebra September 5, 23
Complementation Eample Find complement of F = + F = ( + ) DeMorgan s F = ( ) () DeMorgan s F = ( + )( + ) Reduction -> eliminate double negation on F = (+ )( + ) This format is called product of sums ENIN2 L6: More Boolean Algebra September 5, 23
ENIN2 L6: More Boolean Algebra September 5, 23 Conversion Between Canonical Forms Eas to convert between minterm and materm representations For materm representation, select rows with s = + + = m 7 + m 6 + m 3 = Σ(3, 6, 7) = M M M 2 M 4 M 5 = Π(,,2,4,5) = (++)(++ )(+ +)( ++)( ++ )
Representation of Circuits All logic epressions can be represented in 2- level format Circuits can be reduced to minimal 2-level representation Sum of products representation most common in industr. ENIN2 L6: More Boolean Algebra September 5, 23
Summar Truth table, circuit, and boolean epression formats are equivalent Eas to translate truth table to SOP and POS representation Boolean algebra rules can be used to reduce circuit sie while maintaining function All logic functions can be made from AND, OR, and NOT Easiest wa to understand: Do eamples! Net time: More logic gates! ENIN2 L6: More Boolean Algebra September 5, 23