Lecture 2

1
Lecture 2 Boolean Algebra

• Lecturer Amy Ching
• Date 21st Oct 2002

2
Binary Systems

• Computer hardware works with binary numbers, but
  binary arithmetic is much older than computers
• Ancient Chinese Civilisation (3000 BC)
• Ancient Greek Civilisation (1000 BC)
• Boolean Algebra (1850)

3
Propositional Logic

• The Ancient Greek philosophers created a system
  to formalise arguments called propositional
  logic.
• A proposition is a statement that could be TRUE
  or FALSE
• Propositions could be compounded into by means of
  the operators AND, OR and NOT

4
Propositional Calculus Example

• Propositions, that may be TRUE or FALSE
• it is raining
• the weather forecast is bad
• A combined proposition
• it is raining OR the weather forecast is bad

5
Propositional Calculus Example

• We can equate propositions, for example by
  writing
• I will take an umbrella it is raining OR the
  weather
• forecast is bad
• or equivalently we can write
• If it is raining OR the weather forecast is bad
• Then I will take an umbrella
• OR
• Rain Bad Weather Forecast Take Umbrella

6
Diagrammatic representation

• We can think of the umbrella proposition as a
  result that we calculate from the weather
  forecast or the fact that it is raining

Rain
Take Umbrella
OR
Bad Weather Forecast
7
Truth Tables

• Since propositions can only take two values, we
  can express all possible outcomes of the umbrella
  proposition by a table

Raining Bad Weather Umbrella
False False False
False True True
True True True
True False True
8
Boolean Algebra

• Propositional logic is too cumbersome to express
  arguments of any complexity.
• An equivalent, more tractable system of logic was
  introduced by the English mathematician Boole in
  1850.

9
Boolean Algebra

• A Boolean variable has only one of two values
  true or false (1 or 0), called logic values.
• A Boolean variable can be a function of other
  Boolean variables, i.e. Z F(A, B, C, D).
• We can also express the function in terms of a
  Truth Table
• A Truth Table is a tabulated list contains a
  clear relationship between all possible
  combination of input variables and the resultant
  operation.

10
Fundamental OperatorsAnd Operator

• Three fundamental operators AND, OR and NOT.
• AND Operator
• Z A ? B
• The AND operation is represented by the symbol
  ?. The truth table or logic table of the AND
  operation is as follows

11
Fundamental Operators OR Operator

• OR Operator
• Z A B
• The OR operation is represented by the
  symbol. Note that the OR operation is not
  related to addition in ordinary arithmetic. The
  truth table for OR is as follows

12
Fundamental Operators NOT Operator

• NOT Operator
• or Z A
• The NOT operation is designated by an overline or
  an hyphen.
• In words, the above expression is Z is equal to
  a NOT. The truth table for the NOT operation is
  as follows
• The NOT operation is a complement operation.

13
Fundamentals of Boolean Algebra

• The truth values are replaced by 1 and 0
• 1 TRUE
• 0 FALSE
• Operators are replaced by symbols
• ' NOT
• OR
• AND

14
Precedence

• Further simplification is introduced by
  introducing a precedence for the evaluation of
  the operators.
• (The highest precedence operator is evaluated
  first.)

Operator Symbol Precedence
NOT ' Highest
AND Middle
OR Lowest
15
All outcomes can be written as
NOT '
AND Operator ()
OR Operator ()
16
Boolean Algebra Laws

• 1) Communicative laws 2) Associative laws
• A B B A A(BC) (AB)C
• AB BA (AB)C A(BC)
• 3) Distributive laws 4) Absorption Law
• A? (BC) (A ? B) (A ? C) A? (AB) A
  (A ? B)
• 5) Complement Law
• A 1
• A ? 0
• Other useful relationship
• 1) A ? 1 A 2) A ? 0 0 3) A 1 1 4) A 0
  A
• 5) A A A 6) A ? A A

17
DeMorgans Theorem

• 1)
• 2)
• Both forms of the DeMorgans Theorem have
  complement of an entire expression, and the
  effect of this complementing is to interchange
  each to a ? and each ? to a and to
  complement each variable
• Expression 1) is also described as inputs A and B
  with a NAND operator
• Expression 2) is also described as inputs A and B
  with a NOR operator

18
Simplification of Boolean Equation Using
DeMorgans Theorem

• Simplify Y (AB) ? (AC)
• Y (AB) ? (AC)
• A ? A A ? C B ? A B ? C
• A A ? C A ? B B ? C
• A ? (1CB) B ? C Redundance Law
• A B ? C

19
Sum of Product (SOP) and Product of Sum (POS)

• Product term - is a single variable of the logic
  product of several variables. The variables may
  or may not be complemented. e.g. XYZ, Y
• Sum term - is a single variable or the sum of
  several variables. The variable may or may not be
  complemented e.g. XY,
• Sum of products expression - is a product term of
  several product terms logically added together
  e.g.
• Product of sums expression - is a sum term or
  several sum terms logically multiplied together
  e.g.

20
Conversion of a truth table into SOP and POS

• Sum of product solution
• Product of sum solution

21
Derivation of SOP and POS

• Sum of Products expression (Minterm Form)
• 1) From a truth table
• 2) The product terms from each row in which the
  output is a 1 are collected
• 3) The desired expression is the sum of these
  products e.g.
• Product of Sums expression (Maxterm Form)
• 1) Form a truth table
• 2) Construct a column to contain the sum terms
• 3) The required expression is the product of sums
  terms from the row in which the output is 0
  e.g.

22
Karnaugh Map (K-Map)

• The Karnaugh map provides a formal systematic
  approach to the problem of minimisation of logic
  functions. e.g.
• In the Karnaugh map, every possible combination
  of the binary input variables is represented on
  the map by a square ( or cell).
• For N input variables, we have 2n square.

23
Layout of Karnaugh Map
24
Use of K-Map

• In this way, by inspection, it is obvious that
  terms can be combined and simplified using the
  theorem. e.g.
• To plot the SOP function on Karnaugh map, a 1
  is entered in each square corresponding to a
  product term in the function.

25
Use of K-Map

• To use the map to form the POS function, a 0 is
  entered in each cell corresponding to each sum
  term in the function. Result of simplification
  should then be in POS form.

26
Representation of Karnaugh Map

• Truth Table vs Karnaugh Map

Truth Table
Karnaugh Map
27
Use of K-Map

• There is a correspondence between top and bottom
  rows, and between extreme left and right-hand
  columns.

28
Simplification using a K-Map

• Simplify
• Solution

29
Example 1
30
Example 2
31
Example 3
32
Example 4
33
Example 5