1. Boolean Algebra

2. Summary Digital Circuits (Chips/ICs/Transistors) Basic Logic gates
Boolean Algebra, Basic Rules and Identities
Logic Simplification using Boolean Algebra
De Morgan’s Theorem and its Application in Logic Simplification
Karnaugh Maps – Logic simplification using K’Maps
Implicants

3. Chips/ICs Our world is full of integrated circuits (ICs)
We can found ICs starting from Microprocessor in our computer to almost every modern electrical device such as Car, TV, CD Player, Cell Phone, Electric oven, washing machine etc.
Made from different electrical components such as transistors, resistors, capacitors and diodes

4. Gordon E. Moores’s Laws
Transistors per an IC: Doubling of the number of transistors on integrated circuits every two years (at least for 1 more decade)
Cost per transistor: As the size of transistors has decreased, the cost per transistor has decreased as well
Computing performance per unit cost: As the size of transistors shrinks, the speed at which they operate increases

5. Transistor count is the most common measure of chip complexity.
Processor
Transistor count
Date of introduction
Manufacturer
Intel 4004
2300
1971
Intel
Intel 8008
2500
1972
Intel 8080
4500
1974
Intel 8088
29 000
1979
Intel 80286
134 000
1982
Intel 80386
275 000
1985
Intel 80486
1 200 000
1989
Pentium
3 100 000
1993
AMD K5
4 300 000
1996
AMD
Pentium II
7 500 000
1997
AMD K6
8 800 000
Pentium III
9 500 000
1999
AMD K6-III
21 300 000
AMD K7
22 000 000
Pentium 4
42 000 000
2000
Barton
54 300 000
2003
AMD K8
105 900 000
Itanium 2
220 000 000
Itanium 2 with 9MB cache
592 000 000
2004
Cell
241 000 000
2006
Sony/IBM/Toshiba
Core 2 Duo
291 000 000
Core 2 Quad
582 000 000
G80
681 000 000
NVIDIA
POWER6
789 000 000
2007
IBM
Dual-Core Itanium 2
1 700 000 000
Quad-Core Itanium Tukwila[1]
2 000 000 000
2008

6. Boolean algebra
There are only two possible values for any quantity and for any arithmetic operation 1 or 0
It does not matter how many or few terms we add together, either.

7. Addition in Boolean Algebra
Is not same as real-number algebra

8. Multiplication in Boolean Algebra
Is same as in real-number algebra. Anything multiplied by 0 is 0, and anything multiplied by 1 remains unchanged

9. Logic Gates
A logic gate is an elementary building block of a digital circuit.
There are AND, OR, NOT, NAND, NOR, EXOR and EXNOR gates.
Most logic gates have two inputs and one output.
At any given moment, every terminal is in one of the two binary conditions low (0) or high (1), represented by different voltage levels.
The logic state of a terminal can, and generally does, change often, as the circuit processes data.
In most logic gates, the low state is approximately zero volts (0 V), while the high state is approximately five volts positive (+5 V).

10. Boolean Addition corresponds to the logical function of an "OR" gate

11. Boolean addition corresponds to the logical function of an “AND" gate

12. Boolean compliment corresponds to the logical function of a “NOT" gate

13. Boolean Identities
The sum of anything and zero is the same as the original "anything."
This identity is no different from its real-number algebraic equivalent

14. Boolean Identities The sum of anything and one is one
Different from normal algebra

15. Boolean Identities Adding A and A together
Is same as connecting both inputs of an OR gate to each other and activating them with the same signal

16. Boolean Identities
The sum of a variable and its complement is 1

17. Boolean Identities
Just as there are four Boolean additive identities (A+0, A+1, A+A, and A+A'), so there are also four multiplicative identities: Ax0, Ax1, AxA, and AxA'. Of these, the first two are no different from their equivalent expressions in regular algebra:

18. Boolean Identities
The third multiplicative identity: The product of a Boolean quantity and itself is the original quantity,
since 0 x 0 = 0 and 1 x 1 = 1

19. Boolean Identities
The fourth multiplicative identity: The product of a variable and its complement is 0

20. Boolean Identities (Summary)

21. Boolean Identities
Double complement: a variable inverted twice. Complementing a variable twice (or any even number of times) results in the original Boolean value.

22. Laws of Boolean Algebra
The commutative law/property tells that, we can reverse the order of variables that are either added together or multiplied together without changing the truth of the expression

23. Laws of Boolean Algebra
Associative law tells that, we can associate groups of added or multiplied variables together with parentheses without altering the truth of the equations

24. Laws of Boolean Algebra
Distributive property: The Boolean expression formed by the product of a sum, and in reverse shows how terms may be factored out of Boolean sums-of-products

25. Basic Boolean Algebraic properties
Commutative, Associative, and Distributive

26. Boolean Rules
Boolean algebra finds its most practical use in the simplification of logic circuits.
If we translate a logic circuit's function into symbolic (Boolean) form, and apply certain algebraic rules to the resulting equation to reduce the number of terms and/or arithmetic operations, the simplified equation may be translated back into circuit form for a logic circuit performing the same function with fewer components.
If a equivalent function may be achieved with fewer components, the result will be increased reliability and decreased cost of manufacture.

27. Boolean Rules
This rule may be proven symbolically by factoring an "A" out of the two terms, then applying the rules of A + 1 = 1 and 1A = A to achieve the final result:

28. Boolean Rules

29. Boolean Rules
Proving using truth table

30. Boolean Rules
Simplification of a product-of-sums expression

31. Boolean Rules (Summary)

32. DeMorgan's Theorems

33. DeMorgan's Theorems
Reducing the expression (A + (BC)')' to A’BC using DeMorgan's Theorems

34. DeMorgan's Theorems
Reducing the expression (A + (BC)')' to A’BC using DeMorgan's Theorems

35. DeMorgan's Theorems
Maintaining the grouping implied by the complementation bars for the expression is crucial to obtaining the correct answer

36. DeMorgan's Theorems
Applying the principles of DeMorgan's theorems to the simplification of a gate circuit
Label the outputs of the first NOR gate and the NAND gate
Finally, write an expression (or pair of expressions) for the last NOR gate

37. DeMorgan's Theorems
Reduce the expression using the identities, properties, rules, and theorems (DeMorgan's) of Boolean algebra

38. DeMorgan's Theorems (Review)
DeMorgan's Theorems describe the equivalence between gates with inverted inputs and gates with inverted outputs. Simply put, a NAND gate is equivalent to a Negative-OR gate, and a NOR gate is equivalent to a Negative-AND gate.
When "breaking" a complementation bar in a Boolean expression, the operation directly underneath the break (addition or multiplication) reverses, and the broken bar pieces remain over the respective terms.
It is often easier to approach a problem by breaking the longest (uppermost) bar before breaking any bars under it. You must never attempt to break two bars in one step!
Complementation bars function as grouping symbols. Therefore, when a bar is broken, the terms underneath it must remain grouped. Parentheses may be placed around these grouped terms as a help to avoid changing precedence.

39. Karnaugh Maps
Applying Boolean algebra can be awkward in order to simplify expressions
It is laborious and requires remembering all the laws
The Karnaugh map provides a simple and straight-forward method of minimizing Boolean expressions
With the Karnaugh map Boolean expressions having up to four and even six variables can be simplified.
Karnaugh map provides a pictorial method of grouping together expressions with common factors and therefore eliminating unwanted variables.
Karnaugh map can also be described as a truth table.

40. Karnaugh Maps
Minterm: (Standard product or canonic product term) such as AB’CD or A’BCD’ etc. where each variable used once and once only.
Maxterm: (Standard sum or canonical sum term) such as (A+B’+C+D) or (A’+B+C+D’) where each variable used once and once only
Sum of products: (Minterm canonic form or canonic sum function
f(A,B,C,D)=AB’CD+A’BCD’+A’BC’D
Product of sums: (Maxterm canonic form or canonic product function
f(A,B,C,D)=(A+B’+C+D) (A’+B+C+D’)(A’+B+C’+D)
Adjacent Cells: If two occupied cells of a Karnaugh are adjacent, horizontally, vertically (but not diagonally) then one variable is redundant. Adjacent cells differ in the value of only one variable.
(Rule of adjacency - can knock off one variable as A+A’=1)

41. Karnaugh Maps
Combining all adjacent 1’s more than once doesn’t matter unless no 1 is left out, as A + A = A and A.A = A
Physical, Logical adjacency

42. Karnaugh Maps
The correspondence between the Karnaugh map and the truth table
(two variable)
The values inside the squares are copied from the output column of the truth table, therefore there is one square in the map for every row in the truth table

43. Karnaugh Maps Consider the following map. The function plotted is:
Z = f(A,B) = A B’+ AB
Referring to the map the two 1’s are grouped together. The variable B has its true and false form within the group. Eliminate B leaving only A which only has its true form
Using algebric simplication:
Z = A   + AB
Z = A(   + B)
Z = A

44. Karnaugh Maps
Consider the expression z = f(A,B) = A’B’+AB’+A’B plotted on Karnaugh map:
The first group labeled I, consists of two 1s which correspond to A = 0, B = 0 and A = 1, B = 0. Put in another way, all squares in this example that correspond to the area of the map where B = 0 contains 1s, independent of the value of A. So when B = 0 the output is 1. The expression of the output will contain the term
For group labeled II corresponds to the area of the map where A = 0. The group can therefore be defined as . This implies that when A = 0 the output is 1. The output is therefore 1 whenever B = 0 and A = 0 Hence the simplified answer is Z = A’ + B’

45. Karnaugh Maps Given the truth table
The Boolean algebraic expression is
m = a'bc + ab'c + abc' + abc.
Minimization is done as follows.
m = a'bc + abc + ab'c + abc + abc' + abc
= (a' + a)bc + a(b' + b)c + ab(c' + c)
= bc + ac + ab
bc + ac + ab

46. Karnaugh Maps The Karnaugh map for 4 variables
q = a'bc'd + a'bcd + abc'd' + abc'd + abcd + abcd' + ab'cd + ab'cd'
q = bd + ac + ab
This expression requires 3 2-input and gates and 1 3-input or gate.
RULE: Minimization is achieved by drawing the smallest possible number of circles, each containing the largest possible number of 1s.

47. Karnaugh Maps Imlpicant: Prime Imlpicant: Essential Prime Imlpicant:
Each of the terms i.e product terms that are combined to become sum of products later on are called implicants
Prime Imlpicant:
Largest possible group of values. For that group we can not find larger group
An implicant can get submerged in to a prime implicant.
Essential Prime Imlpicant:
At least one 1 or cell, which not been covered in any other group should be covered
Non Essential Prime Imlpicant:
One way of combining 1s which is not covered otherwise as a an essential prime implicant

48. Gordon E. Moore
Each year computer chips become more powerful yet cheaper than the year before. Gordon Moore, one of the early integrated circuit pioneers and founders of Intel once said,
"If the auto industry advanced as rapidly as the semiconductor industry, a Rolls Royce would get a half a million miles per gallon, and it would be cheaper to throw it away than to park it.”