Post on 11-Jan-2016
Lecture 2 – Boolean Algebra
Lecturer: Amy ChingDate: 21st Oct 2002
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)
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
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
Propositional Calculus Example
We can equate propositions, for example by writing:I will take an umbrella = it is raining OR the weatherforecast is bad
or equivalently we can write:If it is raining OR the weather forecast is badThen I will take an umbrella
ORRain Bad Weather Forecast Take Umbrella
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
Bad Weather Forecast
OR Take Umbrella
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
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.
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.
Fundamental OperatorsAnd OperatorThree fundamental operators AND, OR and NOT.
AND OperatorZ = A B
The AND operation is represented by the symbol “”. The truth table or logic table of the AND operation is as follows:
A B Z = A B
0 0 0
0 1 0
1 0 0
1 1 1
Fundamental Operators –OR OperatorOR 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:
A B Z = A B
0 0 0
0 1 1
1 0 1
1 1 1
Fundamental Operators –NOT OperatorNOT 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.
AZ
A
0 1
1 0
AZ
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
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
All outcomes can be written as:
A B Z = A B
0 0 0
0 1 0
1 0 0
1 1 1
A B Z = A B
0 0 0
0 1 1
1 0 1
1 1 1
A
0 1
1 0
AND Operator (•) OR Operator (+) NOT
'
Boolean Algebra Laws1) Communicative laws 2) Associative laws
A + B = B + A A+(B+C) = (A+B)+C AB = BA (AB)C = A(BC)3) Distributive laws 4) Absorption Law
A (B+C) = (A B) + (A C) A (A+B) = A +(A B)5) Complement Law
A + = 1A = 0
Other useful relationship:1) A 1 = A 2) A 0 = 0 3) A + 1 = 1 4) A + 0 = A5) A + A = A 6) A A = A
A
A
DeMorgan’s Theorem1) 2)
Both forms of the DeMorgan’s 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
BABA )(
BABA )(
Simplification of Boolean Equation Using DeMorgan’s Theorem
Simplify Y = (A+B) (A+C)Y = (A+B) (A+C) = A A + A C + B A + B C = A + A C + A B + B C = A (1+C+B) + B C – Redundance Law = A + B C
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. X+Y,
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.
X
X
XYZXYXYX ,,
ZYXX )(,
Conversion of a truth table into SOP and POS
X Y Z ProductTerms
SumTerms
0 0 1
0 1 0
1 0 1
1 1 1
Sum of product solution Product of sum solution
YX
YXYX
XY
YX
YX YX YX
XYYXYXZ
)( YXZ
Derivation of SOP and POSSum of Products expression (Minterm Form)1) From a truth table2) The product terms from each row in which the
output is a “1” are collected3) The desired expression is the sum of these
products e.g.
Product of Sums expression (Maxterm Form)1) Form a truth table2) Construct a column to contain the sum terms3) The required expression is the product of sums
terms from the row in which the output is “0” e.g.
XYYXYXZ
)( YXZ
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.
BABBA
Layout of Karnaugh Map
XY
X X
Y
Y
XY
Z
0 1
0
1
00
1
0
101101
AB
CD00 101101
00
01
11
10
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.
BABBA
0A
B1
0
1
Y=AB
1
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.
00XY
Z01
0
1
A=(X+Y+Z).(X+Y+Z)
0
11 10
0
Representation of Karnaugh Map Truth Table vs Karnaugh Map
A B Z = A B
0 0 0
0 1 1
1 0 1
1 1 1
0A
B1
0
1
1
11
0
Truth Table Karnaugh Map
Use of K-Map There is a correspondence between top and
bottom rows, and between extreme left and right-hand columns.
00XY
Z01
0
1
A=YZ
1
11 10
1
Simplification using a K-Map Simplify
Solution
CACABCABAY
00AB
C01
0
1
1
11 10
1 1
1 1
CACACBY
Example 1
Example 2
Example 3
Example 4
Example 5