Introdução à Álgebra Booleana: Teoremas e Aplicações, Resumos de Direito Digital

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Chapter 2

Algebraic Methods for the Analysis and

Synthesis of Logic Circuits

Fundamentals of Boolean Algebra (1)

• (^) Basic Postulates
• (^) Postulate 1 (Definition) : A Boolean algebra is a closed algebraic system containing a set K of two or more elements and the two operators  and +.
• (^) Postulate 2 (Existence of 1 and 0 element) : (a) a + 0 = a (identity for +), (b) a 1 = a (identity for )
• (^) Postulate 3 (Commutativity) : (a) a + b = b + a , (b) a  b = b  a
• (^) Postulate 4 (Associativity) : (a) a + ( b + c ) = ( a + b ) + c (b) a  ( b  c ) = ( a  b)  c
• (^) Postulate 5 (Distributivity) : (a) a + ( b  c ) = ( a + b ) ( a + c ) (b) a  ( b + c ) = a  b + a  c
• (^) Postulate 6 (Existence of complement) : (a) (b)
• (^) Normally is omitted. a  a  1 a  a  0

Fundamentals of Boolean Algebra (3)

• (^) Theorem 4 (Absorption) (a) a + ab = a (b) a ( a + b ) = a
• (^) Examples :
  • (^) ( X + Y ) + ( X + Y ) Z = X + Y [T4(a)]
  • (^) AB '( AB ' + B ' C ) = AB ' [T4(b)]
• (^) Theorem 5 (a) a + a ' b = a + b (b) a ( a ' + b ) = ab
• (^) Examples :
  • (^) B + AB ' C ' D = B + AC ' D [T5(a)]
  • (^) ( X + Y )(( X + Y )' + Z ) = ( X + Y ) Z [T5(b)]

Fundamentals of Boolean Algebra (4)

• (^) Theorem 6 (a) ab + ab ' = a (b) ( a + b )( a + b ') = a
• (^) Examples :
  • (^) ABC + AB ' C = AC [T6(a)]
  • (^) ( W ' + X ' + Y ' + Z ')( W ' + X ' + Y ' + Z )( W ' + X ' + Y + Z ')( W ' + X ' + Y + Z ) = ( W ' + X ' + Y ')( W ' + X ' + Y + Z ')( W ' + X ' + Y + Z ) [T6(b)] = ( W ' + X ' + Y ')( W ' + X ' + Y ) [T6(b)] = ( W ' + X ') [T6(b)]

Fundamentals of Boolean Algebra (6)

• (^) Theorem 8 (DeMorgan's Theorem) (a) ( a + b )' = a ' b ' (b) ( ab )' = a ' + b '
• (^) Generalized DeMorgan's Theorem (a) ( a + b + … z )' = a ' b ' … z ' (b) ( ab … z )' = a ' + b ' + … z '
• (^) Examples :
  • (^) ( a + bc )' = ( a + ( bc ))' = a '( bc )' [T8(a)] = a '( b ' + c ') [T8(b)] = a ' b ' + a ' c ' [P5(b)]
  • (^) Note: ( a + bc )'  a ' b ' + c '

Fundamentals of Boolean Algebra (7)

• (^) More Examples for DeMorgan's Theorem
  • (^) ( a ( b + z ( x + a ')))' = a ' + ( b + z ( x + a '))' [T8(b)] = a ' + b ' ( z ( x + a '))' [T8(a)] = a ' + b ' ( z ' + ( x + a ')') [T8(b)] = a ' + b ' ( z ' + x '( a ')') [T8(a)] = a ' + b ' ( z ' + x ' a ) [T3] = a ' + b ' ( z ' + x ') [T5(a)]
  • (^) ( a ( b + c ) + a ' b )' = ( ab + ac + a ' b )' [P5(b)] = ( b + ac )' [T6(a)] = b '( ac )' [T8(a)] = b '( a ' + c ') [T8(b)]

Switching Functions

• (^) Switching algebra : Boolean algebra with the set of elements K = {0, 1}
• (^) If there are n variables, we can define switching functions.
• (^) Sixteen functions of two variables (Table 2.3):
• (^) A switching function can be represented by a table as above, or by a switching expression as follows:
• (^) f 0 ( A , B ) = 0, f 6 ( A , B ) = AB ' + A ' B , f 11 ( A , B ) = AB + A ' B + A ' B ' = A ' + B , ...
• (^) Value of a function can be obtained by plugging in the values of all variables: The value of f 6 when A = 1 and B = 0 is: = 0 + 1 = 1. 2 2 n AB f 0 f 1 f 2 f 3 f 4 f 5 f 6 f 7 f 8 f 9 f 10 f 11 f 12 f 13 f 14 f 15 00 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 01 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 10 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 11 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1  0 ' 1 '  0

Truth Tables (1)

• (^) Shows the value of a function for all possible input combinations.
• (^) Truth tables for OR, AND, and NOT (Table 2.4): ab f ( a , b ) =a+b ab f ( a , b ) =ab a f ( a ) =a ' 00 0 00 0 0 1 01 1 01 0 1 0 10 1 10 0 11 1 11 1

Algebraic Forms of Switching Functions (1)

• (^) Literal : A variable, complemented or uncomplemented.
• (^) Product term : A literal or literals ANDed together.
• (^) Sum term : A literal or literals ORed together.
• (^) SOP (Sum of Products) :
• (^) ORing product terms
• (^) f ( A , B , C ) = ABC + A ' C + B ' C
• (^) POS (Product of Sums)
• (^) ANDing sum terms
• (^) f ( A , B , C ) = ( A ' + B ' + C ')( A + C ')( B + C ')

Algebraic Forms of Switching Functions (2)

• (^) A minterm is a product term in which all the variables appear exactly once either complemented or uncomplemented.
• (^) Canonical Sum of Products ( canonical SOP ) :
  • (^) Represented as a sum of minterms only.
  • (^) Example : f 1 ( A , B , C ) = A ' BC ' + ABC ' + A ' BC + ABC (2.1)
• (^) Minterms of three variables: Minterm Minterm Code Minterm Number A ' B ' C ' 000 m 0 A ' B ' C 001 m 1 A ' BC ' 010 m 2 A ' BC 011 m 3 AB ' C ' 100 m 4 AB ' C 101 m 5 AB C' 110 m 6 ABC 111 m 7

Algebraic Forms of Switching Functions (4)

• (^) Example : Given f ( A , B , Q , Z ) = A ' B ' Q ' Z ' + A ' B ' Q ' Z + A ' BQZ ' + A ' BQZ , express f ( A , B , Q , Z ) and f '( A , B , Q , Z ) in minterm list form. f ( A , B , Q , Z ) = A ' B ' Q ' Z ' + A ' B ' Q ' Z + A ' BQZ ' + A ' BQZ = m 0 + m 1 + m 6 + m 7 =  m (0, 1, 6, 7) f '( A , B , Q , Z ) = m 2 + m 3 + m 4 + m 5 + m 8 + m 9 + m 10 + m 11 + m 12 + m 13 + m 14 + m 15 =  m (2, 3, 4, 5, 8, 9, 10, 11, 12, 13, 14, 15)
• (^) (2.6)
• (^) AB + ( AB )' = 1 and AB + A ' + B ' = 1, but AB + A ' B '  1. mi i n     0 2 1 1

Algebraic Forms of Switching Functions (5)

• (^) A maxterm is a sum term in which all the variables appear exactly once either complemented or uncomplemented.
• (^) Canonical Product of Sums ( canonical POS ) :
  • (^) Represented as a product of maxterms only.
  • (^) Example : f 2 ( A , B , C ) = ( A + B+C )( A+B+C ')( A '+ B+C )( A '+ B + C ') (2.7)
• (^) Maxterms of three variables: Maxterm Maxterm Code Maxterm Number A + B + C 000 M 0 A + B + C ' 001 M 1 A + B ' +C 010 M 2 A + B '+ C ' 011 M 3 A '+ B + C 100 M 4 A '+ B + C ' 101 M 5 A '+ B '+C 110 M 6 A '+ B '+ C ' 111 M 7

Chapter 2 19

Algebraic Forms of Switching Functions (7)

• (^) Truth tables of f 1 ( A,B,C ) of Eq. (2.3) and f 2 ( A,B,C ) of Eq. (2.7) are identical.
• (^) Hence, f 1 ( A , B , C ) =  m (2,3,6,7) = f 2 ( A,B,C ) =  M (0,1,4,5) (2.10)
• (^) Example: Given f ( A,B,C ) = ( A+B+C ')( A+B '+ C ')( A ' +B+C ')( A ' +B ' +C '), construct the truth table and express in both maxterm and minterm form. - (^) f ( A,B,C ) = M 1 M 3 M 5 M 7 =  M (1,3,5,7) =  m (0,2,4,6) Row No. ( i ) Inputs ABC Outputs f(A,B,C)= M(1,3,5,7) = m(0,2,4,6) 0 000 1 m 0 1 001 0   M 1 2 010 1  m 2 3 011 0   M 3 4 100 1 m 4 5 101 0   M 5 6 110 1  m 6 7 111 0  ^ S. Isrie, MSc. M 7

Algebraic Forms of Switching Functions (8)

• (^) Relationship between minterm mi and maxterm Mi :
  • (^) For f ( A,B,C ), ( m 1 )' = ( A'B'C )' = A + B + C ' = M 1
  • (^) In general, ( mi )' = Mi (2.11) ( Mi )' = (( mi )')' = mi (2.12)