1. International Journal of Electronics and Communication Engineering & Technology (IJECET), ISSN 0976 –
INTERNATIONAL JOURNAL OF ELECTRONICS AND
6464(Print), ISSN 0976 – 6472(Online) Volume 4, Issue 5, September – October (2013), © IAEME
COMMUNICATION ENGINEERING & TECHNOLOGY (IJECET)
ISSN 0976 – 6464(Print)
ISSN 0976 – 6472(Online)
Volume 4, Issue 5, September – October, 2013, pp. 225-236
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IJECET
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ON THE ALGEBRAIC RELATIONSHIP BETWEEN BOOLEAN FUNCTIONS
AND THEIR PROPERTIES IN KARNAUGH-MAP
Binoy Nambiar1, Jigisha Patel2
1
2
Electronics and Communications Engineering Department, SVNIT, Ichchanath, Surat, India.
Electronics and Communications Engineering Department, SVNIT, Ichchanath, Surat, India.
ABSTRACT
The differential algebra for Boolean function has already been developed, and the areas of
applications are still being explored. This paper uses the concept of the differential maximum
function to understand the implications and the behaviour of Boolean function .The paper focuses on
the behaviour of a Boolean function in a Karnaugh-map plot, and built a theoretical framework
for more development in this area. New form of the decompositions (AND bi-decomposition,
OR bi-decomposition and the EX-OR bi-decomposition) are proposed and their meanings and
implications are explored in a way that can simplify their applications.
The developed mathematics can also be extended to include the new forms’ implication to the
hyperspace cube, and also to include the Boolean function vector derivatives. The explanations of the
derivatives, both scalar and vector, is given using the K-map and the hyperspace cubes, and can be
directly applied to algorithm development of analysing different combinations of minterms.
Key words— Boolean Function Expansion, Karnaugh-Map, Differential Maximum, Boolean
Function Vector Derivative, Boolean Function.
1. INTRODUCTION
In recent times, there have been new developments in the area of application of Karnaugh
maps and its property of minimizing Boolean functions, e.g. in generation of error-detection codes
[1], and in database management [2] and management itself. Boolean function expansions are
already used in many applications, including cryptography e.g. to devise Boolean functions with
maximum algebraic immunity [5]. Also there have been attempts to develop Karnaugh-maps for
quantum logic [3] [4]. The concept of Boolean differential equations has also been developed and is
widely used in analysis, synthesis and testing of digital circuits [8] [9] [10]. The Karnaugh-map is an
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efficient way of minimizing function, but too much prone to error if the number of variables
increases. This paper redevelops the concepts of bi-decompositions and total maximum of a Boolean
function by including the concepts into Karnaugh-Map.
The paper is organized as follows
Section II presents the necessary terms and definitions to understand the forthcoming theorems.
Section III presents the theorem with its proof. Section IV develops the algebra for the theorem and
the practical use of the theorem. Section V discusses and compares the developments according to
the present situation.
2. DEFINITIONS AND TERMINOLOGIES
Here the maximum of a differential is called as the delta derivative, for reasons which will be
clear in further course of paper.
The definition of delta derivative of a Boolean function used in this paper as
For any Boolean function F(x1, x2,…., xi,…xn), its delta derivative with respect to xi is
∆F
= F(x1, x2,…., 0,…xn) + F(x1, x2,…., 1,…xn)
∆xi
(I)
Here, the values 1 and 0 are taken for variable xi.
All the arguments/terms/definitions presented from here onwards are with reference to K-map only,
unless otherwise stated.
Neighbour: A neighbour of a cell is the cell adjacent to a given cell.
Table: 1 A 3-variable Karnaugh map plot
1
1
0
1
1
3
2
5
7
6
1
4
For e.g., In Table: 1, the cells numbered 0, 5and 3 are adjacent to cell number 1. Hence, cells 0, 5
and 3 are neighbours to cell 1.
Region: A region is group of cells. It should have more than one cell.
Address: It is the cell number. In this figure, the numbers which are at the bottom right corner are the
respective addresses. The address indicates the state of the Boolean variables in decimal. For e.g. the
cell having address 0 has value 1; which shows that when A = 0, B = 0 and C = 0, the function’s
value is 1.
A filled cell indicates the presence of a 1 in the cell, and an empty cell indicates the presence of
value 0, or absence of value 1.
In the above presented example, cells 0, 1, 3 and five are filled cells, whereas cells 2, 4, 6 and 7 are
empty cells.
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A neighbourhood of a cell is the collection of all adjacent cells of the given cell. Example,
here, the neighbourhood of cell 1 is (0, 3, 5). From now onwards, every cell will be represented by
its minterm number, and while considering the value of a function theoretically, without referring to
K-map, then it shall be represented as mi. For e.g. the cell 1 will be represented as m1 and so on.
A sub-neighbourhood is the subset of neighbourhood. In the above example, cells 0 and 3 are
neighbours of cell 1, but they are not the only neighbours. Hence they shall be represented as subneighbourhood. Thus, for the presented example, set of cells (0, 3) is a sub-neighbourhood of the cell
1. In minterm form, the set of cells (0, 3) shall be represented as (m0, m3). The concept of filled and
empty is also applicable to neighbourhood and sub-neighbourhood
A connected region: It is a region that has its every filled cell with at least one neighbour
which is a filled cell, or a filled neighbour, or every filled cell in a connected region has at least 1
filled neighbour. A function having such a plot in K-map shall be called a connected function.
Simply connected region: A simply connected region is a connected region that satisfies the
following condition: If all the neighbours of a given cell are filled, then the given cell must also be
filled.
Table: 2 A 4-variable Karnaugh map plot
1
1
0
1
3
2
5
7
6
15
14
1
4
1
12
13
1
8
9
1
11
10
In Table: 2, two connected regions are shown, both of them are not connected with each other
in any sense. Here, it can be said that both the regions are isolated regions with respect to each other,
and the complete set of both the connected regions can be said as set of connected regions.
Isolated cell: An isolated cell is a cell having no filled neighbours. For e.g. In Table: 3, the cells 1, 2,
7 are all isolated cells
Table: 3 A 3-variable Karnaugh map plot
1
0
1
1
3
2
7
6
1
4
5
Isolation function: The function depicting the isolated cells is called the isolation function. For e.g.,
the cell 7 is an isolated cell, hence the isolation function can be represented on a K-map as shown in
table 5. It can be denoted as Fi.
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Table: 4 A 3-variable Karnaugh map plot
1
1
0
1
1
3
2
1
4
5
7
6
0
1
3
2
7
6
1
4
5
Hole: A hole is an empty cell (cell with value 0), which has filled neighbours.
For e.g., in Table: 4, the cell 3 is having value 0, but all its neighbours (cells 1, 5, 7) have value 1.
Thus cell 3(m3) is a hole.
Hole function: The function which gives value 1 at holes of a function is called hole function of a
function F. For e.g. the hole function of the function shown in table no. 4 can be shown as shown in
table no. 6.
Table: 6 A 3-variable Karnaugh map plot
1
0
1
3
2
4
5
7
6
It can be denoted as Fh.
Vector: The usual definition of vector is assumed, which is a set of variables. E.g. (A, B) is a vector,
being a set of variables.
Dimension of vector: The no. of variables in a vector. E.g. vector (A, B) has dimensionality of 2, or
the given vector is 2-dimensional.
Nth neighbour: The cell which is n-1 cells away from a reference cell is nth neighbour, or which is at
a Hamming distance of 2. For e.g. in Table 8, the cell 7 is 2nd neighbour of cell 0.
Nth neighbour hole: An empty cell having all its nth neighbours filled. For e.g. in Table 8, the cell 1 is
a 1st neighbour hole as well as 2nd neighbour hole.
Nth neighbour isolated cell: A filled cell having all its nth neighbours empty. For e.g. in Table 9, the
cell 1 is a 2nd neighbour isolation cell, but not 1st.
Table: 7 A 4-variable Karnaugh map plot
1
0
1
3
2
1
4
5
7
6
12
13
15
1
8
9
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14
1
11
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3. THEOREM
In this section, we use the notations developed earlier and derive the AND bi-decomposition.
Theorem 1: Any set of simply connected regions in a Karnaugh-map can be expressed as
F=
∆F
∏ ∆x
i
(II)
i
where F is a Boolean function of the variables with respect to which the K-map is plotted, for any
Boolean function of n-variables F(x1, x2,…., xi,…xn).
3.1. Proof
The proof presented here is for 3-variable Boolean function, but the proof can be easily
extended to inculcate any no. of variables.
Assume a simply connected function F(A, B, C).
Considering the definition of delta derivative of a Boolean function F(A, B, C) with respect to C as
∆F
= F(A, B, 0) + F(A, B, 1)
∆C
The minterms m0 and m1 can be represented as m0 = F(0, 0, 0) i.e. the value of function at (0, 0, 0)
state, m1 = F(0, 0, 1) and so on
∆F ∆F ∆F
= [F(A, B, 0) + F(A, B, 1)][F(0, B, C) + F(1, B, C)][F(A, 0, C) + F(A, 1, C)]
∆A ∆B ∆C
= [A’B’{F(0, 0, 0) + F(0, 0, 1)} + A’B{F(0, 1, 0) + F(0, 1, 1)} + AB’{F(1, 0, 0) + F(1, 0, 1)} +
AB{F(1, 1, 0) + F(1, 1, 1)}][B’C’{F(0, 0, 0) + F(1, 0, 0)} + B’C{F(0, 0, 1) + F(1, 0, 1)} + BC’{F(0,
1, 0) + F(1, 1, 0)} + BC{F(0, 1, 1) + F(1, 1, 1)}][A’C’{F(0, 0, 0) + F(0, 1, 0)} + A’C{F(0, 0, 1) +
F(0, 1, 1)} + AC’{F(1, 0, 0) + F(1, 1, 0)} + AC{F(1, 0, 1) + F(1, 1, 1)}]
(III)
Eq. (III) can be simplified as
= A’B’C’(m0 + m1m2m4) + A’B’C(m1 + m0m3m5) + A’BC’(m2 + m0m3m6) + A’BC(m3 + m1m2m7)
+ AB’C’(m4 + m0m5m6) + AB’C(m5 + m1m4m7) + ABC’(m6 + m7m2m4) + ABC(m7 + m3m5m6)
(IV)
Now, according to the definition of a simply connected region, if any cells’ all neighbours are
filled, then the given cell has to be necessarily filled. For example, let’s assume the neighbourhood
of cell m0. The neighbours are m1, m2 and m4. If m1, m2 and m4, all 3 are filled then necessarily m0
has to be filled, that means the first term of A’.B’.C’ will be included as a product term. Thus for
each term, the argument can be extended, thus proving the theorem for 3-variables, which, as it can
be clearly seen, can be extended to n-variable Boolean function.
From here onwards, the expansion in the 1st theorem presented in eq. (II) will be called as delta
expansion, and the result of the expansion will be called F∆ (F delta). So, if a function F is a simply
connected function, then
F = F∆
(V)
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This also shows that although a 4 variable K-map can be plotted in a 2x2 matrix or 3x1
matrix, if we want the adjacencies to be evident, then it should be plotted in 2x2. This can be
extended to n-variables. If an n-variable K-map is to be plotted in 2 dimensions, then the no. of
variables n should be split into 2 numbers n1 and n2 in such a way that the product of a and b should
be maximum, and this can again be extended for a plot in m-dimensions. The number of variables n
should be divided into m numbers n1, n2, … ni, … nm such that
m
∑ n = n,
i =1
(VI)
i
such that their product is maximum i.e.
∏ n is maximum
i
(VII)
i
The 1st theorem can further be extended to include 2nd and 3rd delta derivatives as well i.e. the
function F(x1, x2,…..xi, …..xn) can be expanded as
F(x1, x2,…..xi, …..xn) =
∆2F
∏∏ ∆xi∆xj
i j≠ i
(VIII)
And so on. To differentiate between the expansions made using 1st delta derivative and 2nd delta
derivative, ‘order’ of expansion shall be used. Order, in this article, is defined as the order of the
derivative used in the expansion. For e.g. the form of expansion shown in equation (VIII) has order =
2. The maximum order possible is N-2, for any Boolean function having N number of variables.
4. FURTHER DEVELOPMENT PROPOSED
Consider a Boolean function
F (A, B, C) = B’C’ + AB’ + A’BC
(IX)
Here, 1st order expansion shall be used. Now the corresponding delta derivatives are
∆F
= B + C’
∆A
∆F
= A’ + B’
∆C
(XI)
∆F
=1
∆B
Thus,
(X)
(XII)
∆F ∆F ∆F
= B’ + A’C = F∆
∆A ∆B ∆C
(XIII)
This function’s plot in K-map is
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A= 0
A=1
B = 0, C = 0
1
0
1
4
Table 8: Plot of F∆
B = 0, C = 1 B = 1, C = 1
1
1
1
3
1
5
7
B = 1, C = 0
2
6
Whereas the original function’s plot is
A=0
A=1
Table 9: Original function’s plot
B = 0, C = 0 B = 0, C = 1 B = 1, C = 1
1
1
0
1
3
1
1
4
5
7
B = 1, C = 0
2
6
Thus, we can see that when the condition of a function F being a simply connected region is
not satisfied, in other words, if a function has holes in its plot, then, when the function is expanded as
a product of its delta derivatives, the holes are filled, and the function becomes a simply connected
region. This is due to the fact that every term in minterm expansion had its neighbours, i.e. for e.g.,
in the expansion in eq. (IV) if we take any term, lets say A’B’C’(m0 + m1m2m4), the term is not m0
as it should be in a normal expansion , but (m0 + m1m2m4). This means that if the cell 0 (m0) is a
hole, then it will be filled, as its neighbours m1, m2 and m4 are filled, the term A’B’C’ in the
expansion will have minterm value 1.
There is another interesting transformation occurring in the previous example. The original
function can be represented as
F = A’( B ⊕ C' ) + AB’
The delta function can be represented as
F∆
= A’(B + C’) + AB’
We can see that the ex-or has been removed and has been replaced by or. The ‘ex-or’ function
always produces a hole, as the ‘ex-or’ function, when plotted, has an empty cell between two filled
cells. This hole is filled by the delta function. This is an interesting property of the delta function.
Now, according to the definition of the hole function, every hole has its neighbours filled, but the
hole itself empty. So the conditions to be satisfied are
1) The selected cell (hole) must be empty.
2) All its neighbours must be filled
Now, the hole function can be written as
Fh = F’. F∆
(XV)
Proof: On minterm expansion of right hand side of the equation, the left side of the equation is
directly obtained.
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Similarly, for the isolation function, the conditions obeyed by the minterms are
1) The cell should be filled.
2) Its neighbours should be empty.
Using a similar analysis as used in the hole, the isolation function can be written as
Fi = F. F∆
(XVI)
Where F'∆ is the delta function of F’.
Thus it can be directly checked whether a function is a simply connected function or not, simply by
checking whether its hole function is zero or not.
Hence, the condition to be satisfied by a function for being simply connected is
Fh = F' . F∆ = 0
(XVII)
Similarly, the set of connected regions in a function F can also be defined in terms of function and
the delta function.
Each minterm in the connected region satisfies the following two conditions.
1) The cell must be filled
2) Each filled cell must have at least one filled neighbour.
Thus the set of connected regions in a function F (from here onwards it will be called the continuity
function) can be written as
Fc = F'∆ '
(XVIII)
Thus now we have all the components of a function
[1] Hole function (Holes): Fh = F' . F∆
[2] Isolation function (Isolated cells): Fi = F. F'∆
[3] Set of connected regions in a function F (function excluding isolated cells): Fc = F'∆ '
The delta function doesn’t have the holes present in the function F. The function can thus be
recovered by removing the holes from the delta function. Using this fact, we can present the second
theorem as
Theorem 2: Any Boolean function F can be represented as
F = F∆ ⊕ F'F∆
(XIX)
The set of connected regions excludes the isolated regions from the function F. Thus we can recover
the function F by adding the isolated cells to the set of connected regions. Thus the third theorem can
be presented as
Theorem 3: Any Boolean function can be represented as
'
F = F'∆ + FF'∆
(XX)
Thus a Boolean function can be represented in terms of 2 sets of two functions each, i.e. one set of
(F ∆ , FF ∆ ) and the other set (F' , FF' ) . Thus a function F has been broken into three parts, a continuos
part, holes and isolated cells. Here, it is interesting to note that the isolation function can directly be
obtained from the hole function by placing an F’ (or putting the complement) in place of the function
∆'
∆
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F. This is because holes in F are the isolated cells in F’, and vice versa. This can be proved by
finding the isolated in F’, the result of which will be:
The isolation function for F’ = the hole function of F.
Now, the relation of the delta derivative with the general derivative will be derived.
The definition of a derivative is
∂F
= F(A, B, C) ⊕ F(A’, B, C)
∂A
(XXI)
On doing a similar expansion as done to obtain eq. (II), we get
∂F ∂F ∂ F
= A’B’C’(m0’m1m2m4 + m0m1’m2’m4’) + A’B’C(m1’m0m3m5 + m1m0’m3’m5’) + …..
∂A ∂ B ∂ C
and so on. The term can be seen as: The function will have the value one when any cell is either a
hole or an isolated cell. This can be understood from analysis of any one of the minterms in product.
For e.g. in 1st term A’B’C’, the product term is (m0’m1m2m4 + m0m1’m2’m4’). The 1st term
(m0’m1m2m4) is one only when m0 = 1 and m1 = m2 = m4 = 0, where m1, m2 and m4 are the
neighbours of m0. Thus the cell m0 should be one whereas its neighbours must be zero, which means
that it must be an isolated cell (or the cell must be in isolation). Similarly, on seeing the other term,
we can say that it will be one only when the cell m0 = 0, but its neighbours must be one, which is a
hole. Thus we can say that the product of all derivatives of a function is the ‘or’ of hole function and
isolation function. Thus
∂F
(XXII)
∏ ∂xi = F' F∆ + F F'∆
i
The relation between delta derivative and the general derivative is
∆F
∂F
=F+
∆x i
∂x i
(XXIII)
Proof: On minterm expansion of right hand side of the equation, the left side of the equation is
directly obtained.
Hence F ∆ can be written as
(XXIV)
F ∆ = F + ∏ ∂F
i ∂x i
The equations (XXV) and (XXVI) are very important as they provide the transformation between the
delta derivative and the general derivative.
Now, the reverse conversion (delta derivative to derivative) is
∂F ∆F ∆F'
=
∂x i ∆ x i ∆ x i
(XXV)
Proof: On minterm expansion of right hand side of the equation, the left side of the equation is
directly obtained.
Thus, if we multiply all possible derivatives of a function (only 1st derivatives), the result is
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∂F
∏ ∂x i = F∆ F'∆
i
(XXVI)
Hence, if a function is devoid of any holes, then
∂F
∏ ∂x i
i
= F F'∆
(XXVII)
Which is precisely the isolation function, and similarly if a function is devoid of any isolated cells,
∂F
∏ ∂x i
i
= F' F ∆
(XXIX)
Which is the hole function.
Now,
D x (F1F2 ) = D x (F1 )D x (F2 )(F2 + F'1 + D'x (F'1 ))(F1 + F'2 + D'x (F'2 )) (XXXI)
i
i
i
i
i
Proof: On minterm expansion of right hand side of the equation, the left side of the equation is
directly obtained.
Thus, on taking the product of all possible delta derivatives (1st delta derivatives)
(F1F2 ) ∆ = (F1 ) ∆ (F2 + F'1 + ∏ D'x (F'1 ))(F2 ) ∆ (F1 + F'2 + ∏ D'x (F'2 )) (XXXII)
i
i
i
i
Now, the above made developments shall be extended to include the vector derivatives, and explore
the results.
In the same manner as done for delta derivative with respect to one derivative, the delta derivative
for group of variables (vector) is defined as
DF
= F(A, B, C) + F(A', B', C)
DX
(XXXIII)
Where the group of variables (vector) = X is assumed as X = (A, B)
When the minterm expansion is done for the above made choice of vector, result is
DF
= A’B’C’(m0 + m6) + ….. and so on
DX
(XXXIV)
When the product of all such vector delta derivatives having n. of variables in the vectors = 2, is
taken, the result is
DF
∏ DX = A’B’C’(m0 + m3m5m6) + ….. and so on (XXXV)
i
i
Here, the minterms in addition to the m0 the terms are m3, m5 and m6, which are the cells
which are the second cell from the cell m0. This is because of the no. of variables in the vector is 2,
or the vectors with respect to which the derivatives are taken, are 2-dimensional, hence the
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neighbours which are 1 cell away from the reference cell are analysed, or 2nd neighbours are
analysed. Thus this result is useful if any cell is a 2nd neighbour hole or not. Extending the argument,
the product of all the delta derivatives done with respect to vectors of dimensionality n, will be the
function itself if and only if the function has no nth neighbour holes.
If product of all the delta derivatives of function F, with the maximum dimension as n, is
taken, then, the product will definitely be F, if and only if the function has no 1st, 2nd…nth holes.
This product is assumed to be nth dimensional (n-D) delta function ( Fn∆ ) i.e.
F n∆ = ∏
i
DF
DX i
(XXXVI)
where i covers all the possible vectors till dimension n.
When the maximum dimension of the delta derivatives reaches one more than the total
number of variables upon which the function is depending, then the product returns the function F.
All the relations between the delta derivatives and the general derivatives are applicable to vector
derivatives as well.
5. DISCUSSION AND CONCLUSION
As we can see, the delta-derivative, or the logical maximum, of a Boolean function, is very
much useful in analysing a function, its behaviour in Karnaugh-map, and to analyse each cell (A
Boolean state) and its neighbourhood, showing that Boolean differential algebra very much similar
to the real valued differential algebra, that relies on the concept of neighbourhood of a real point.
This paper indicates that the Karnaugh-map along with the properties of the Boolean differential
algebra can be also used for different applications. Although here, the formulas are developed by
taking into consideration all the neighbours, 2nd neighbours etc. it is done only to keep the theory
symmetric with respect to all the variables. The expansion presented here can also be done for some
of the variables/group of variables (vectors), which will then analyse the neighbours only in the
direction of the change of variables selected, thus a path can be created, on the Karnaugh-Map to be
analysed. The required variables should be taken in the derivatives, and the order of derivative
should be decided on the basis of the Hamming distance from the reference cell. The papers [1] and
[11] are one of the few examples that give a method to use K-map in error-correction and detection,
which is very intuitive and detection and correction are almost on an equal footing in the methods
shown. The derivatives and the expansions derived here can be used for the implementation of the
techniques shown in [1] and [11]. Thus any technique which uses the concept of Hamming distance,
even over a specific pattern, e. g. Genetic algorithm, will find the mathematics developed here
useful.
6. APPENDIX
Appendix – I
Proof for Fd’c’ = Fc’d’ (This proof can be extended for any permutation between any variables)
Fd’ = F(A,B,C) = A’B’[C’{F(0,0,0,0) + F(0,0,0,1)} +
C{F(0,0,1,0) + F(0,0,1,1)}]
+ A’B[C’{F(0,0,0,0) + F(0,0,0,1)} + C{F(0,0,1,0) + F(0,0,1,1)}] +
AB’[C’{F(0,0,0,0) + F(0,0,0,1)} + C{F(0,0,1,0) + F(0,0,1,1)}] + AB[C’{F(0,0,0,0) +
F(0,0,0,1)} + C{F(0,0,1,0) + F(0,0,1,1)}]
(A 1.1)
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12. International Journal of Electronics and Communication Engineering & Technology (IJECET), ISSN 0976 –
6464(Print), ISSN 0976 – 6472(Online) Volume 4, Issue 5, September – October (2013), © IAEME
F(A,B) = Fd’c’ = A’B’[F(0,0,0,0) + F(0,0,0,1) + F(0,0,1,0) + F(0,0,1,1)] + A’B[F(0,0,0,0) +
F(0,0,0,1) + F(0,0,1,0) + F(0,0,1,1)] + AB’[F(0,0,0,0) + F(0,0,0,1) + F(0,0,1,0) +
F(0,0,1,1)] + AB[F(0,0,0,0) + F(0,0,0,1) + F(0,0,1,0) + F(0,0,1,1)]
(A 1.2)
Similar procedure for Fc’d’ will also yield the same result. Hence it’s proved that Fc’d’ = Fd’c’.
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