Organizational%20Remarks

1
Organizational Remarks

• I dont use the exercises in Mano/Kime this way
  you can use them as review exercises (for many of
  them there are solutions on the web
  www.prenhall.com/mano).
• On the 8th of october there will lab sessions
  instead of the lecture on the 10th of october

2
Remarks

• Twos complement
• For the math minded, otherwise ignore
• Boolean ring is not a Boolean algebra
• given a Boolean ring you can equip it with a
  Boolean algebra structure (and conversely)
• a ? b ab (AND)
• a ? b ab ab (OR)
• a 1 a (NOT)

3
Remarks

• For the math minded, otherwise ignore
• given a Boolean algebra you can equip it with a
  Boolean ring structure
• xy x ? y,
• x y (x ? y) ? (x ? y).
• (? AND, ?OR, NOT the given operations of the
  Boolean algebra)

4
Logic diagrams, Boolean expressions, and
Truth(1/0) Tables

• LDs ??BEs ?? TTs

5
Logic diagrams (LDs), Boolean Expressions (BEs),
and Truth Tables (TTs) (or 1/0-tables)

• The object which is lurking behind the scenes is
  of the course the notion of a Boolean logic
  function (or for short a Boolean function) a
  Boolean function of n variables is a mapping
  from the set 0,1n to the set 0,1. (What do
  you mean by 0,1n ? )
• Of course, a Boolean function is completely
  determined by a Truth table (1/0-table)

6
LDs, BEs, and TTs

• LDs ??BEs ?? TTs
• Each Boolean function has a unique TT
• In general a Boolean function or a TT can have
  more than one LD (or for that matter BE)
  associated to it.
• For any two of (LD, BE, TT) can readily go back
  and forth

7
LDs, BEs, and TTs

• For any two of (LD, BE, TT) can readily go back
  and forth
• From TT to BE for each 1 in the output introduce
  an appropriate minterm (a product term which
• contains each of
  the variables or their negation),
  then S is the
• sum of these minterms

A B Cin
0 0 0
0 0 1
0 1 0
0 1 1
1 0 0
1 0 1
1 1 0
1 1 1
S
0
1
1
0
1
0
0
1
ABC
ABC
ABC
ABC
S ABC ABC ABC ABC
8
LDs, BEs, and TTs BE ? TT

• S ABC ABC ABC ABC
• How do you get the Truth Table (1/0-table)? Etc.
• We have done also the transition from TT to LD.
  Given the LD it is not difficult to come up with
  the TT.
• It is also not difficult to imagine how you would
  go directly from LD to BE and vice versa.
• Make sure that you are conversant with all
  possible transitions

9
Reduction (Simplification) of Boolean Expressions

• It is usually possible to simplify the
  canonical SOP (or POS) forms.
• A smaller Boolean equation generally
  translates to a lower gate count in the target
  circuit.
• We cover three methods algebraic reduction,
  Karnaugh map reduction, and tabular
  (Quine-McCluskey) reduction.

10
Reduced Majority Function Circuit

• Compared with the AND-OR circuit for the
  unreduced majority function, the inverter for C
  has been eliminated, one AND gate has been
  eliminated, and one AND gate has only two inputs
  instead of three inputs. Can the function by
  reduced further? How do we go about it?

11
The Algebraic Method

• Consider the majority function, F. We apply
  the algebraic method to reduce F to its minimal
  two-level form

12
An Aside from Alfred Tarski

• What did you mean in high school by the following
  statement (ab)(a-b) a2 b2?
• As such it can be confusing it is not an
  equation but an identity the following is much
  better
• For all a in R, for all b in R (ab)(a-b) a2
  b2 R stands for the real numbers.
• And so it goes when we say in Boolean algebra
  context aaa. We mean For all a in B0, 1 aa
  a

13
An Aside from Alfred Tarski

1. And so it goes when we say in Boolean algebra
   context aaa. We mean For all a in B01, aa
   a
2. Or another example
3. abcabcabcabc bcacab (pertains to the
   majority function)
4. Let B0,1.
5. For all a in B, for all b in B, for all c in B
   abcabcabcabc bcacab
6. Or say 3) is shorthand for 5)

14
The Algebraic Method

• This majority circuit is functionally
  equivalent to the previous majority circuit, but
  this one is in its minimal two-level form

15
Karnaugh Maps Venn Diagram Representation of
Majority Function

• Each distinct region in the Universe
  represents a minterm.
• This diagram can be transformed into a
  Karnaugh Map.

16
K-Map for Majority Function

• Place a 1 in each cell that corresponds to
  that minterm.
• Cells on the outer edge of the map wrap
  around

A
C
B
17
Adjacency Groupings for Majority Function
Slightly different bookkeeping (no conceptual
change)

1

• F BC AC AB

18
Minimized AND-OR Majority Circuit

• F BC AC AB
• The K-map approach yields the same minimal
  two-level form as the algebraic approach.

19
K-Map Groupings

• Minimal grouping is on the left, non-minimal
  (but logically equivalent) grouping is on the
  right.
• To obtain minimal grouping, create smallest
  groups first.

B
A
D
C
20
K-Map Corners are Logically Adjacent
Slightly different bookkeeping (no conceptual
change)
1
21
K-Maps and Dont Cares

• There can be more than one minimal grouping, as
  a result of dont cares.

22
Five-Variable K-Map

• Visualize two 4-variable K-maps stacked one on
  top of the other groupings are made in three
  dimensional cubes.

23
Six-Variable K-Map

• Visualize four 4-variable K-maps stacked one on
  top of the other groupings are made in three
  dimensional cubes.