# Thinking Mathematically by Robert Blitzer

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```					Boolean Algebra
Boolean Algebra: Structure
 Instudying propositional logic (true, false),
we learned about certain operators, that can
be viewed as functions f:{T, F}2→{T, F}.

 Forexample, (T, T) = T, (T, F) = T,
(F, T) = T, and (F, F) = F.

 From there,
fundamental properties of wff's:
Boolean Algebra: Structure
   ABBA          ABBA
commutative laws
   (A  B)  C  A  (B  C)
(A  B)  C  A  (B  C)
associative laws
   A  (B  C)  (A  B)  (A  C)
A  (B  C)  (A  B)  (A  C)
distributive laws
   A  F  A A  T  A identity laws
   A  A'  T A  A'  F complement laws
Boolean Algebra: Structure
   We also studied sets, and within Set
Theory we discovered that given sets A,
B and C within a universe U:
Boolean Algebra: Structure
   AB=BA           AB=BA
commutative laws
   (A  B)  C = A  (B  C)
(A  B)  C = A  (B  C)
associative laws
   A  (B  C) = (A  B)  (A  C)
A  (B  C) = (A  B)  (A  C)
distributive laws
   AF=A        AU=A          identity laws
   A  A' = U A  A' =  complement laws
Boolean Algebra: Structure
 One  of the beauties of mathematics is
that there is a lot of consistency and
many recurring patterns.
 Many different models satisfy the same
(or similar) properties.
 Systems like the ones we have just seen
are examples of a model called a
Boolean algebra.
Boolean Algebra: Structure
A Boolean algebra is a set B on which are
defined
   two binary operations: + and 
   and one unary operation '
 and   in which there are two distinct elements
   and 1 (multiplicative identity)
 such  that the following properties hold for
all x, y, z  B:
Boolean Algebra: Structure
   x+y=y+x                   xy=yx
commutative laws
   (x + y) + z = x + (y + z)
(x  y)  z = x  (y  z)
associative laws
   x + (y  z) = (x + y)  (x + z)
x  (y + z) = (x  y) + (x  z)
distributive laws
   x+0=x             x1=x        identity laws
   x + x' = 1       x  x' = 0    complement laws
Boolean Algebra: Structure
   Example:
 Consider the set B = {0, 1} and define the operations +

and  to be x + y = max(x, y) and x  y = min(x, y). Let 0'
= 1 and 1' = 0.
 The above forms a Boolean algebra.

• commutative: obviously max(x,y) = max(y,x)
and the same for min.
• associative: intuitively true.
• distributive: max(x, min(y,z)) =
min(max(x,y), max(x,z))
• identity
• complement
Boolean Algebra: Structure
 The    idempotent property: x + x = x
   x + x = (x + x)  1        multiplicative identity

         = (x + x)  (x + x') additive complement
         = x + (x  x')       distributive property
         =x+0               multiplicative complement
Boolean Algebra: Structure
 The   Principle of Duality:
   If some equality holds within a Boolean
algebra, then the corresponding equality
obtained by substituting
• + for ,
•  for +,
• 0 for 1
• and 1 for 0
   will also be true.
Boolean Algebra: Structure
 Uniqueness:  If x + x1 = 1 and x  x1 = 0,
then x1 = x'.
 De Morgan: (x + y)' = x'  y',
(x  y)' = x' + y'
 Absorption: x + (x  y) = x,
x  (x + y) = x
 Double negation: (x')' = x
Boolean Algebra: Structure
 One   more thing:
   If (B, + , , ') is a Boolean algebra, then |B|
= 2n
Logic Networks
 Claude    Shannon (1938)
   Boolean algebras and logic circuits.
x=1                   x=0

x=1
1
x=0
Logic Networks

A      B      A+B   A      B       A B
A          A'
1      1       1    1      1        1
1          0
1      0       1    1      0        0
0          1
0      1       1    0      1        0
inverter
0      0       0    0      0        0
OR gate             AND gate
Logic Networks

A    B (A + B)'     A    B   (A  B)'
1    1    0         1    1      0
1    0    0         1    0      1
0    1    0         0    1      1
0    0    1         0    0      1
NOR gate            NAND gate
Logic Networks and Truth
Tables
 Consider a truth table involving three
variables.
   There will be 23 = 8 lines.
   There are 2  23 = 28 = 256 unique truth tables.

   Each truth table can be represented by
logic circuits using the basic gates shown
earlier.
Logic Networks and Truth
Tables
   Consider the following table:
x1   x2   x3   f(x1,x2,x3)
1    1    1         1
1    1    0         0
1    0    1        1
1    0    0        1
0    1    1        0
0    1    0        0
0    0    1        1
0    0    0        0
Logic Networks and Truth
Tables
 The truth table on the previous slide
can be implemented by any number
of logic circuits using AND, OR and
inverter gates in combination.
 One such circuit can be built using

the sum-of-products notation.
 Let's go back to that table.
Logic Networks and Truth
Tables
x1   x2   x3   f(x1,x2,x3)
1    1    1        1         x1x2x3      sum of
1    1    0        0                    products
1    0    1        1         x1x'2x3
1    0    0        1         x1x'2x'3
0    1    1        0
0    1    0        0
0    0    1        1         x'1x'2x3
0    0    0        0
x1x2x3 + x1x'2x3 + x1x'2x'3 + x'1x'2x3
Logic Circuits and Truth Tables
x1
x2
x3

x1
x2
x3

x1
x2
x3

x1
x2
x3
Logic Circuits and Truth Tables
 But  is this the only circuit that
corresponds to the earlier truth table?
 Not by a long shot!
 Consider the expression x1x3 + x1x'2 +
x'2x3
Logic Circuits and Truth Tables
x1   x2   x3   x1x3   x1x'2   x'2x3     x 1x 3 +
x1x'2+x'2x3
1    1    1     1      0       0           1
1    1    0     0      0       0           0
1    0    1     1      1       1           1
1    0    0     0      1       0           1
0    1    1     0      0       0           0
0    1    0     0      0       0           0
0    0    1     0      0       1           1
0    0    0     0      0       0           0
Logic Circuits and Truth Tables
 The   truth table on the previous slide is
identical to the earlier one.
 Note that it only requires three AND
gates leading to the OR gate.
 Note that each of these AND gates
requires only 2 (not 3) inputs.
 It's simpler.
 It's cheaper.
Minimization
 There  is a technique for finding the
minimal circuit for a given truth table.
 It involves a special map called the
Karnaugh Map.
 Here are the general pictures for
Karnaugh Maps corresponding to two-
variable, three-variable and four-
variable expressions.
Karnaugh Maps
x1 x'1                 x1x2   x1x'2 x'1x'2 x'1x2
x2      1      1
x3      1     1     1     1
x'2
x'3     1     1           1

x1x2   x1x'2 x'1x'2 x'1x2
x3x4
x3x'4
x'3x'4
x'3x4
Karnaugh Maps
 Note that adjacent squares have only one
variable that changes (e.g. – x1 to x'1)
 Note that Karnaugh Maps "wrap around",
top-to-bottom, left-to-right.
 We look for sub-rectangles (perhaps
wrapping around) with both the number of
rows and the number of columns being
powers of two (1, 2, 4, etc.)
Karnaugh Maps
 Find  the smallest number of largest
rectangles that cover all the 1's and
only the 1's.
 Each rectangle is a product of
variables (or inverses of variables).
 The simplest expression is the sum of
these rectangles.
Karnaugh Maps
 Consider    the following map:
x1x2     x1x'2  x'1x'2       x'1x2
x3x4         1         1            1
x3x'4         1         1
x'3x'4                  1
x'3x4

x1x3 + x'2x3x4 + x1x'2x'4
Karnaugh Maps
 Consider    the following map:
x1x2     x1x'2  x'1x'2       x'1x2
x3x4         1         1            1
x3x'4         1         1
x'3x'4                  1
x'3x4

x1x3 + x'2x3x4 + x1x'2x'4
Karnaugh Maps
 Consider   the following map:
x1x2     x1x'2  x'1x'2       x'1x2
x3x4        1                    1      1
x3x'4
x'3x'4                  1
x'3x4        1                           1

x2x4 + x'1x3x4 + x1x'2x’3x'4
Karnaugh Maps
 Consider      the following map:
x1x2     x1x'2  x'1x'2   x'1x2
x3x4          1        1       1       1
x3x'4          1                        1
x'3x'4         1                        1
x'3x4          1                        1

x2    + x3x4

```
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