Lecture 2 – Boolean Algebra Lecturer: Amy Ching Date: 21 st Oct 2002.
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Transcript of Lecture 2 – Boolean Algebra Lecturer: Amy Ching Date: 21 st Oct 2002.
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Lecture 2 – Boolean Algebra
Lecturer: Amy ChingDate: 21st Oct 2002
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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)
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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
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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
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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
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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
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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
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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.
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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.
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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
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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
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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
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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
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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
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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
'
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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
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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 )(
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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
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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 )(,
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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
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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
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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
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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
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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
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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
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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
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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
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Simplification using a K-Map Simplify
Solution
CACABCABAY
00AB
C01
0
1
1
11 10
1 1
1 1
CACACBY
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Example 1
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Example 2
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Example 3
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Example 4
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Example 5