Boolean Algebra

1
Boolean Algebra
2
Outline

• Laws and theorems of Boolean Algebra
• Switching functions
• Logic functions NOT, AND, OR, NAND, XOR, XNOR
• Switching function representations
• Canonical forms

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Axiomatic of Boolean Algebra

• A Boolean algebra consists of a set B with two
  binary operations ( ? AND, ? OR) and a unary
  operation ( or ? NOT), such that the
  following axioms satisfy
• Set B contains at least two distinct elements a
  and b.
• Closure For every a, b ? B,
• a b ? B
• a ? b ? B
• Commutative Laws For every a, b ? B,
• a b b a
• a ? b b ? a

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Axiomatic of Boolean Algebra

• Associative Laws For every a, b, c ? B,
• (a b) c a (b c)
• (a ? b) ? c a ? (b ? c)
• Identities For every a ? B,
• ? an identity element 0, such that a 0 a
• ? an identity element 1, such that a ? 1 a
• Distributive Laws For every a, b, c ? B,
• a ? (b c) (a ? b) (a ? c)
• Complement For each a ? B, ? an such that
• a 1
• a ? 0

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Boolean function
 
 

• A Boolean function uniquely maps Bn to B.
• A Boolean expression is an algebraic statement
  containing Boolean (binary) variables and
  operators (?, , and ?), that is (AND, OR, and
  NOT)
• A literal is a variable itself or its complement.
  When a Boolean function is implemented with
  logic gates, each literal represents an input to
  a gate, and each term is implemented a gate.

 
 
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Examples

1. F XYZ?
2. F X Y? Z
3. F X? Y? Z X? YZ XZ?
4. Z A? ? B? ? (C D)
5. Z (A? ? (B? ? (C D)))

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Laws and Theorems of Boolean Algebra

• Duality Every Boolean expression is deducible
  from the postulates of Boolean algebra remains
  valid if the operators and the identity elements
  are interchanged. That is interchange OR and AND
  operators and replace 1's by 0's and 0's by 1's.

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Examples

• X 1 1 ? X ? 0 0
• X XY X ? X(X Y)

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Laws and Theorems
Boundness law A 1 1 A ? 0 0
Identity law A 0 A A ? 1 1
Idempotent Theorem A A A A ? A A
Involution Theorem (A?)? A
Theorem of complementarity A A? 1 A ? A? 0
Commutative law A B B A AB BA
Associative law A (B C) (A B) C A(BC) (AB)C
Distributive law A (B C) AB AC A BC (AB)(AC)
DeMorgan's Theorem (A B)? A? B? (AB)? A? B?
Absorption law A AB A A(A B) A
Consensus Theorem ABBCA?C ABA?C (AB)(BC)(A?C) (AB)(A?C)
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Examples

• Simplify the following Boolean expressions to a
  minimum number of literals
• X X? Y ? X Y
• XY X? Y ? Y
• X (X? Y ) ? XY
• X? Y? Z X? YZ XY? ? X? Z XY?
• XY X? Z YZ ? XY X? Z
• (XY)(X?Z)(YZ) ? X? Y XZ

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Switching Functions

• A switching algebra is a Boolean algebra whose
  set B contains only two values 0 and 1.
• A switching function uniquely maps Bn to B.
• f (X, Y, Z) XY X? Z YZ
• If X 0, Y 1, Z 0, then f (X, Y, Z) 0.
• If X 0, Y 1, Z 1, then f (X, Y, Z) 1.

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Truth Tables

• A switching function can be represented as a
  Boolean function or in a tabular form called
  truth table. A truth table is a list of possible
  combinations of inputs that correspond to the
  values of the switching function (output).

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Example

• Truth table of f (X, Y, Z) XY X? Z YZ

X Y Z f
0 0 0 0
0 0 1 1
0 1 0 0
0 1 1 1
1 0 0 0
1 0 1 1
1 1 0 1
1 1 1 1
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Switching Functions

• There are 16 possible switching functions of two
  variables

X Y 0 ? XY? X X? Y Y ? OR NOR Y? ? X? ? NAND 1
0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1
0 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1
1 0 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1
1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
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Canonical and Standard Forms

• Minterms
• Maxterms

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Minterms

• For two binary variables A and B combined with
  an AND operation, the minterms or standard
  products are AB, AB?, A?B, and A?B?. That is,
  two binary variables provide 22 4 possible
  combinations (minterms.) n variables have 2n
  minterms. Each minterm has each variable being
  primed if the corresponding bit of the binary
  number is a 0 and unprimed if a 1.

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Maxterms

• Similarly, two binary variables A and B combined
  with an OR operation, the maxterms or standard
  sums are AB, AB?, A?B, and A?B?. That is,
  two binary variables provide 22 4 possible
  combinations (maxterms.) n variables have 2n
  maxterms. Each maxterm has each variable being
  primed if the corresponding bit of the binary
  number is a 1 and unprimed if a 0.
• A maxterm is the complement of its corresponding
  minterm, and vice versa.

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Boolean function
 
 

• Sum of Products (or Minterms)
• A Boolean function can be expressed as a sum of
  minterms. The minterms whose sum defines the
  Boolean function are those that give the 1's of
  the function in a truth table.
• Product of Sums (or Maxterms)
• A Boolean function can be expressed as a product
  of maxterms. The maxterms whose sum defines the
  Boolean function are those that give the 0's of
  the function in a truth table.

 
 
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Minterms and Maxterms for Three Binary Variables
Input Input Input Minterm Minterm Maxterm Maxterm
X Y Z Term Designation Term Designation
0 0 0 X?Y?Z? m0 XYZ M0
0 0 1 X?Y?Z m1 XYZ? M1
0 1 0 X?YZ? m2 XY?Z M2
0 1 1 X?YZ m3 XY?Z? M3
1 0 0 XY?Z? m4 X?YZ M4
1 0 1 XY?Z m5 X?YZ? M5
1 1 0 XYZ? m6 X?Y?Z M6
1 1 1 XYZ m7 X?Y?Z? M7
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Examples
X Y Z Function F1 Function F2
0 0 0 0 0
0 0 1 1 0
0 1 0 0 0
0 1 1 0 1
1 0 0 1 0
1 0 1 0 1
1 1 0 0 1
1 1 1 1 1
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Examples

• F1 and F2 can be expressed as a sum of products
  as follows
• F1 X?Y?ZXY?Z?XYZ m1 m4 m7
• F2 X?YZXY?ZXYZ? XYZ m3 m5 m6 m7
• F1 and F2 can also be expressed as a product of
  sums as follows
• F1 (XYZ)(XY?Z)(XY?Z?)(X?YZ?)(X?Y?Z)
• M0 M2 M3 M5 M6
• F2 (XYZ)(XYZ?)(XY?Z)(XY?Z?)
• M0 M1 M2 M4

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Notation

• Boolean functions expressed as a sum of products
  or product of sums are said to be in canonical
  form
• A convenient way to express these function is by
  using a short notation, decimal form
• F1(X, Y, Z) ?m(1,4,7)
• F2(X, Y, Z) ?m(3,5,6,7)
• or
• F1(X, Y, Z) ? M(0,2,3,5,6)
• F2(X, Y, Z) ? M(0,1,2,4)

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Standard forms

• A Boolean function is said to be in standard
  form if the function contains one, two or any
  number of literals. For example
• F1 Y?XYX?YZ? or
• F2 X(Y?Z)(X?YZ?W)
• A Boolean function may be expressed in a
  nonstandard form. For example, the function
• F (WXYZ)(W?X?Y?Z?)

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Example 1

• 1. Given the following truth table. Express F in
  a canonical minterms and maxterms.

X Y Z F
0 0 0 0
0 0 1 1
0 1 0 0
0 1 1 0
1 0 0 1
1 0 1 1
1 1 0 1
1 1 1 1
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Example 2

• 2. Design a digital logic circuit that will
  activate an alarm if a door or window is open
  during non-business hours. Assume that

Clock C 0 (non-business hours) 1 (business hours)
Door D 0 (closed) 1 (opened)
Window W 0 (closed) 1 (opened)
Alarm A 0 (off) 1 (on)
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Conversion between canonical form

• To convert from a sum of products to a product of
  sums rewrite the minterm canonical form in a
  shorthand notation then replace the existing term
  numbers by the missing numbers. For example
• F1(X, Y, Z) ?m(1,3,6,7) ? M(0,2,4,5)

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Conversion between canonical form

• To convert from a product of sums to a sum of
  products rewrite the maxterm canonical form in
  a shorthand notation then replace the existing
  term numbers by the missing numbers. For
  example
• F1(X, Y, Z) ? M(0,2,4,5) ?m(1,3,6,7)

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Conversion between canonical form

• To obtain the minterm (or maxterm) canonical form
  of the complement, given the Boolean function in
  a sum of products (or product of sums) form
  list the term numbers that are missing in For
  example
• F(X, Y, Z) ?m(0,2,4,5) ? F?(X, Y, Z) ?m
  (1,3,6,7)
• F(X, Y, Z) ?M(1,3,6,7) ? F?(X, Y, Z)
  ?M(0,2,4,5)

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Don't Care Conditions

1. F(A,B,C,D) ?m(1,3,7,11,13,15) d(0,2,5)
2. F(A,B,C,D) ?m(4,5,6,7,8,9,10,13) d(0,7,15)
3. BCD increment by 1 function.

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Logic Functions

• AND Operation Z X ? Y

Inputs Inputs Output
X Y Z
0 0 0
0 1 0
1 0 0
1 1 1
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Logic Functions

• OR Operation Z X Y

Inputs Inputs Output
X Y Z
0 0 0
0 1 1
1 0 1
1 1 1
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Logic Functions

• NOT Operation Z X ?

Inputs Output
X Z
0 1
1 0
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Logic Functions

• NAND Operation Z (X ? Y)?

Inputs Inputs Output
X Y Z
0 0 1
0 1 1
1 0 1
1 1 0
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Logic Functions

• NOR Operation Z (X Y)?

Inputs Inputs Output
X Y Z
0 0 1
0 1 0
1 0 0
1 1 0
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Logic Functions

• XOR Operation Z X ? Y

Inputs Inputs Output
X Y Z
0 0 0
0 1 1
1 0 1
1 1 0
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Logic Functions

• XNOR Operation Z (X ? Y)?

Inputs Inputs Output
X Y Z
0 0 1
0 1 0
1 0 0
1 1 1
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Switching function representations

• There are 3 ways to represent a switching
  function
• Boolean expression
• Truth table
• Logic diagram

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Positive and Negative Logic
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Positive and Negative Logic
Truth Table Truth Table Truth Table Positive Logic Positive Logic Positive Logic Negative Logic Negative Logic Negative Logic
x y z x y z x y z
low low low 0 0 0 1 1 1
low high low 0 1 0 1 0 1
high low low 1 0 0 0 1 1
high high high 1 1 1 0 0 0
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Positive and Negative Logic
Truth Table Truth Table Truth Table Positive Logic Positive Logic Positive Logic Negative Logic Negative Logic Negative Logic
x y z x y z x y z
low low low 0 0 1 1 1 0
low high low 0 1 0 1 0 0
high low low 1 0 0 0 1 0
high high high 1 1 0 0 0 1
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Example

• Example Traffic lights -- to define three signals