BOOLEAN ALGEBRA
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BOOLEAN ALGEBRA
-REVIEW
Boolean Algebra was proposed by George Boole in 1853.
Basically AND,OR NOT can be expressed as Venn Diagrams
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Min Terms and Max Terms
Min Terms are those which occupy
minimum area on Venn Diagram
Max Terms are those which occupy
maximum area on Venn diagram.
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LOGIC GATES
Nand and Nor gates are called Universal gates
as any Boolean function can be realized with the
help of Nand and Nor gates only
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For example, NOT, OR, AND gates can be realized
by only Nand gates.
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SIMPLIFICATION OF
BOOLEAN FUNCTIONS
•
•
•
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Algebraic Method
Tabular Method
K-Map Method
Schienman Method
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ALGEBRAIC METHOD
Simplify using algebraic theorems
Advantage:
First Method based on Boolean Algebra theorems
Disadvantage:
No Suitable algorithm to apply (Trial type of method)
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TABULAR METHOD
Also called Quine McClusky Method
Advantage:
It may work for any no. of variables
Disadvantage:
Simplification from table is quite involved
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K-MAP METHOD
Karnaugh Map. Also called Vietch
Karnaugh Method.
Advantage:
Simplest and Widely accepted
Disadvantage:
Applicable for only upto Six variables
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SCHIENMAN METHOD
Columnwise writing of minterms as
decimal numbers and their simplification
Advantage:
Very suitable for computerization. Applicable for any
number of variables. Parallel Processing
Disadvantage:
May not result in most simplified answer for some
problems
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Steps for simplification:
• Try to find single one’s
– 2 one’s
– 4 one’s
– 8 one’s
Always see is a higher combination exists. If a
higher combination exists, wait. Be sure that
you have managed the lower combination first.
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SYMMETRIC FUNCTIONS
• DEFINITION
• PROPERTIES
• IDENTIFICATION
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Definition
A switching function of n variables
f(X1,X2….Xn) is called a symmetric (or
totally symmetric), if and only if it is invariant
under any permutation of its variables. It is
partially symmetric in the variables Xi,Xj
where {Xi,Xj} is a subset of {X1,X2…Xn} if
and only if the interchange of the variables
Xi,Xj leaves the function
unchanged.
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EXAMPLES
• f(x,y,z) = x’y’z+xy’z’+x’yz’
If we substitute x = y and y = x
x = z and z = x
y = z and z = y
TOTALLY SYMMETRIC with respect to x,y,z
• f(x,y,z) = x’y’z + xy’z’
is Prettily Symmetric in the variables x and z.
(x = z and z = x)
• f(x,y,z) = z’y’x + zy’x’
is a Symmetric function
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• f(x,y,z) = y’x’z +yx’z’)
is Not a Symmetric function
This function is symmetric w.r.t x and z, but not symmetric
w.r.t x and y. So Partially Symmetric
•f(x1,x2,x3) = x1’x2’x3’ + x1x2’x3+ x1’x2x3
is not symmetric w.r.t. the variables x1,x2,x3, but is
symmetric w.r.t the variables x1,x2,x3’
>> f is not invariant under an interchange of variables
x1,x3.
That is, x3’x2’x1’+x3x2’x1 +x3’x2x1 != f
>> But f is invariant under an interchange of variables
x1,x3’
That is, x3x2’x1 + x3’x2’x1 + x3x2x1’ = f
So f is symmetric w.r.t the variables x1,x2 and x3’
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The variables in which a function is
symmetric are called the
VARIABLES OF SYMMETRY
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Necessary and Sufficient condition for
function f(x1,x2….xn) to be symmetric is that
it may be specified by a set of numbers
{a1,a2…ak} where 0<an<n,such that it
assumes the value 1 when and only when ai of
the variables are equal to 1. The numbers in
the set are called the a-numbers
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A Symmetric function is denoted by
Sa1,a2…ak (x1,x2….xn), where S designates the
property of symmetry, the subscripts designate
the a numbers, and (x1,x2….xn) designate the
variables of symmetry.
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IDENTIFICATION
The switching function to be tested for
symmetry is written as a table in which all the
minterms contained in the function are listed
by their binary representation
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For example, the function f(x,y,z) =
(1,2,4,7) is written as shown:
x y z a#
The arithmetic sum of each
0 0 1 1
column in the table is computed
0 1 0 1
written under the column. This
1 0 0 1
sum is referred to as a column1 1 1 3
sum. The number of 1’s in each
2 2 2
row is written in the corresp.
position in column a#. This no. is called
ROW SUM.
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If an n-variable function is symmetric and one
of its row sums is equal to some number a, then,
by definition, there must exist n!/(n-a)!a! rows
which have the same row sum.
If all the rows occur the required number of times,
then all colums sums are identical
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For the example, all column sums equal 2,
and there are two row sums, 1 and 3, that must
be checked for “Sufficient Occurrence”.
>> 3!/(3-1)! = 3 ; 3!/(3-3)! = 1
Both row sums occur the required number of
times. Therefore, the function is symmetric
and can be expressed by S1,3(x,y,z).
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The column sums are 6 4 4 6
Since the column sums are not all the same,
further tests must be performed to determine
if the function is symmetric, and if it is, to
find its variables of symmetry.
The column sums can be made the same by
complementing the columns corresponding
to variables x and y.
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The new column sums are now computed and are
found identical. The row sums are determined
next and entered as a#. Each row sum is tested
by the binomial co-efficient occurrence.
4!/(4-2)! = 6 ; 4! /(4-3)!3! = 4
Since, all row sums occur the required number
of times, the function is symmetric,its variables
of summetry are w,x’,y’,z and its a numbers are 2
and 3. ( f = S2,3(w,x’,y’,z))
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If columns w and z are complimented, instead
of x and y, the table shown below results and \
since all its row sums occur the required no. of
times, f can be written as f = S1,2(w’,x,y,z’)
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The column sums are all identical, but row
sum 2 does not occur six times as required.
One way to overcome this difficulty is by
expanding the function about any one of its
variables
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The function can be expanded about w.
w x y z
0 0 0 0
0 0 1 1
0 1 0 1
a#
0
2
2
Column Sums :
x y z
1 1 2
The column sums can be made by complementing
the columns corresponding to variables x and y.
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x’
1
1
0
y’
1
0
1
z a#
0 2
1 2
1 2
Column Sums:
x y z
2 2 2
Each row is tested by the binomial coefficients
for sufficient occurrence.
3!/(3-2)!2! = 3
Symmetry: S2(x’,y’,z)
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w x y z
1 0 1 0
1 1 0 0
1 1 1 1
2 2 1
x
0
1
1
2
y
1
0
1
2
z’
1
1
0
2
a#
1
1
3
a#
2
2
2
The column sums can be made
the same by complementing
the columns corresponding
to variable z.
Each row is tested by the
binomial coefficients for
sufficient occurrence.
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3!/(3-2)! 2! = 3
Symmetry:
S2(x,y,z’)
So the function f is written as
f=w’S2(x’,y’,z) + wS2(x,y,z’)
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questions?
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