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Boolean Algebra

The section of abstract mathematics concerning two values: true and false.

History

George Boole
introduced Boolean algebra in 1847 as part of his book,
The Mathematical Analysis of Logic
. The approach was then fine-tuned by
William Stanley Jevons
,
Ernst Schröder
, and in the late 19th century. The system was applied to circuitry in the 1930’s when
Claude Shannon
, a Mathematician and Electronics Engineer, made the realization that Boolean algebra could be used as tool to analyze logic gates. Any system using sentential logic will also have a way of representation in Boolean Algebra.
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George Boole introduced Boolean algebra in 1847 as part of his book, The Mathematical Analysis of Logic. The approach was then fine-tuned by William Stanley Jevons and Ernst Schröder in the late 19th century. The system was applied to circuitry in the 1930’s when Claude Shannon, a Mathematician and Electronics Engineer, made the realization that Boolean algebra could be used as tool to analyze logic gates. Any system using sentential logic will also have a way of representation in Boolean Algebra.

Rules of Boolean Algebra

There are 13 Basic laws of Boolean algebra.
Rules
"A+1=1"
"A+0=A"
"A*1=A"
"A*0=0"
"A+A=A"
"A*A=A"
"-(-A)=A"
"A+-A=1"
"A*-A=0"
"A+B=B+A"
"A*B=B*A"
"-(A+B)=-A+-B"
"-(A*B)=-A*-B"
These concepts can be easily be represented using Mathematica’s built-in functions.
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Rule 1: A || 1 == 1
In[12]:=
True||1//Boole
Out[12]=
1
◼
Shouldn’t this rule be input as follows (parentheses are required because of operator precedence)?
In[10]:=
(||True)True
Out[10]=
True
◼
In StandardForm you can write this as (using
or
to generate the ∨ operator)
In[23]:=
(∨True)True
Out[23]=
True
◼
I’ve used

(
scA
) instead of
A
as using capital letters (such as
C
,
D
,
K
) can cause problems (reserved words in the system)
◼
Alternatively, use TraditionalForm (Cell > Convert To > TraditionalForm)
In[24]:=
(∨True)True
Out[24]=
True
In[10]:=
False||1
In[20]:=
1||1//FullSimplify//Boole
Out[20]=
1
In[21]:=
0||1//FullSimplify//Boole
Out[21]=
1
◼
Rule 2: A || 0 == A
In[18]:=
1||0//FullSimplify//Boole
Out[18]=
1
◼
Here’s another way to write Rule 2
In[16]:=
(∨False)
Out[16]=
True
In[19]:=
0||0//FullSimplify//Boole
Out[19]=
0
◼
Rule 3: A && 1 == A

In[25]:=
1&&1//FullSimplify//Boole
Out[25]=
1
In[26]:=
0&&1//FullSimplify//Boole
Out[26]=
0
◼
Rule 4: A && 0 == 0

In[27]:=
1&&0//FullSimplify//Boole
Out[27]=
0
In[28]:=
0&&0//FullSimplify//Boole
Out[28]=
0
◼
Rule 5: A || A = A

In[29]:=
1||1//FullSimplify//Boole
Out[29]=
1
In[30]:=
0||0//FullSimplify//Boole
Out[30]=
0
◼
Rule 6: A && A = A

In[31]:=
1&&1//FullSimplify//Boole
Out[31]=
1
In[32]:=
0&&0//FullSimplify//Boole
Out[32]=
0
◼
Rule 7: !(!A) = A

In[33]:=
!(!A)
Out[33]=
A
In[1]:=
(¬(¬a))a
Out[1]=
True
◼
Rule 8: A || !A = 1

In[2]:=
1||!1//FullSimplify//Boole
Out[2]=
1
◼
Boole is not required here
In[22]:=
1||!1//FullSimplify
Out[22]=
True
◼
An alternative:
In[19]:=
(∧¬)False//FullSimplify
Out[19]=
True
◼
Rule 9: A && !A = 0

In[6]:=
1&&!1//FullSimplify//Boole
Out[6]=
0
In[7]:=
0&&!0//FullSimplify//Boole
Out[7]=
0
◼
Rule 10: A || B == B || A
In[10]:=
A||BB||A
Out[10]=
True
◼
Rule 11: A && B == B && A
In[18]:=
(A&&B)(B&&A)//FullSimplify
Out[18]=
(A&&B)(B&&A)
◼
Rule 12:-(A||B) = !A && !B
In[21]:=
(A⊽B)(!A&&!B)
Out[21]=
(A⊽B)(!A&&!B)

Applications

Proficiency in Boolean Algebra is a required skill for all Electrical Engineering. As Shannon observed, any piece of circuitry can be described in terms of Boolean algebra. An example of a use-case will be shown below.
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Typically, the problem or device that is to be solved will be put into a logical project description. From this, an engineer would then create a logical description of the problem. An example of this is shown below.
(!A&&!B&&!C&&!D)||(!A&&!B&&!C&&D)||(A&&B&&C&&D)||(A&&B&&C&&!D)
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Replacing
{A,B,C,D}
by
{,ℬ,,}
and converting this inert expression to TraditionalForm and then converting the Cell to an equation using Format > Style > DisplayFormula:
In[26]:=
(¬∧¬ℬ∧¬∧¬)∨(¬∧¬ℬ∧¬∧)∨(∧ℬ∧∧)∨(∧ℬ∧∧¬)
Equations like these are unique for every problem presented to the engineer, along with their method of approach.
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Using the rules of Boolean Algebra, BooleanConvert simplifies the expressing above:
In[24]:=
BooleanConvert[(!A&&!B&&!C&&!D)||(!A&&!B&&!C&&D)||(A&&B&&C&&D)||(A&&B&&C&&!D)]
BooleanConvert can also convert the expression into different forms.
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BooleanConvert converts above expression to all NAND, AND/NOT, and OR/NOT:
Once the equation is in it’s simplest form, the engineer can then determine which logic gates are required and how to link them correctly.
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Using * as AND gates and + as OR gates, the diagram for the above equation is shown below:
This is the basic process of breaking down a complex issue into it’s simple logic using Boolean algebra.
Further Explorations
Circuit Analysis
Karnaugh Map
Logic
Authorship information
Patrick Griffin
Friday, June 23, 2017
pgrfn5@gmail.com