Babylon University
The college of science for girls
Logic Design
Experiment (2)
Rules and laws of Boolean Algebra
Aim:- To prove the rules of laws of Boolean algebra.
Theory:- As in other areas of mathematics, there are certain well developed rules
And laws that must followed in order to properly apply Boolean algebra.
The most important of these are presented in table (1).
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1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
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X+0 = x
X + 1 = 1
X + x = x
X + x` = 1
X`` = x
X + y = y + x
(x + y) + z = x + (y +z ) = x + y + z
X ( y + z ) = xy + xz
Xy + xy` = x
X + xy = x
X ( x` + y ) = xy
( x + y ) ( x + z ) = x + yz
X + y + z + ……= x . y . z …..
Xy + yz + x`z = xy + x`z
(x + z ) (x` + y ) = xy + x`
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1D
2D
3D
4D
6D
7D
8D
9D
10 D
11 D
12 D
13 D
14 D
15 D
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X .1 = x
X .0 = 0
X . x = x
X . x` = 0
X . y = y . x
(x . y ) . z = x . (y . z ) = x .y .z
X + yz = (x + y ) (x+z )
( x + y ) ( x + y` ) = x
X ( x + y ) = x
X + ( x`y ) = x + y
Xy + xz = x ( y + z )
X . y . z …… = x + y + z …..
( x + y ) (y + z ) ( x + z ) = (x+y) (x`+z)
Xz + x`y = (x + y ) (x` + z )
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Table (1):-Laws of the terms of Boolean algebra.
Note:-
1)Each variable in table (1) could have one of two values (1 or 0).
2)The symbol (+) means OR operation.
The symbol (.) means AND operation.
The symbol (-) over variables means NOT operation.
3) The letter (D) represents the dual of an algebraic expression
The duality principle states that every algebric expression deducible from the postulates of Boolean algebraic remains valid if the operators and the identity elements are interchanged
If the dual of an algebraic expression is
(x+y) . (x+z) = x+yz
Desired , we simply interchange OR & AND operators . and replace 1 S by 0 S by 1 S
Example :- the expression
X+X` = 1 X . X = 0
Every rule can be proved theoretically and practically
Example :-
Lets take rule (14) as an example and try to prove it theoretically
Xy + yz + x`z = xy + yz ( x+x`) + x`z ……. (14)
Taking the left side of the equation
Xy + yz + x`z = xy + yz (x+x`) + x`z
Since
(x+x`) = 1 according to the rule (4) , there is no harm of applying it to the expression as we did in the above expression .
Xy + yz (x+x`) + x`z = xy + xyz + x`yz + x`z
According to the rule (8) .(distributed law )
Xy + xyz + x`yz + x`z = xy (1+z) +x`z`(1+y)
According to rule (2)
1+z=1 , 1+y =1 therefore
Xy (1+z) + x`z (1+y) = xy + x`z = the right side
In this experiment we learn how to prove these previous rules practically
Procedure :-
1) prove rule (8),(8D) respectivically , draw the circuits and state the truth table
2) Repeat procedure (1) to prove rule (11) , (11D) respectively
3) Repeat procedure to prove rule (13),(13D)
Notes :-
In order to prove the rules in the procedures above , we should connect the circuits for the expressions of each side of the rule, then put the results in truth tables.
If the truth tables are identical for both sides of the rule then the rule is proved .
Discussion :-
1)prove the following
a) (x + y ) (x + y`) = x
b) (x + z ) (x` + y ) = xy + x`z
2)Use the rules shown in table (1) to simplify the following
a) (x+y) (x` + y`)
b) Xyz + x`y + xyz`
REFERENCES :-
- Alladin uwaied , " Lectures of digital design "
- Thomas L.Floyed, " Digital Fundamantals"
Ahmed M. Shhaab
المادة المعروضة اعلاه هي مدخل الى المحاضرة المرفوعة بواسطة استاذ(ة) المادة . وقد تبدو لك غير متكاملة . حيث يضع استاذ المادة في بعض الاحيان فقط الجزء الاول من المحاضرة من اجل الاطلاع على ما ستقوم بتحميله لاحقا . في نظام التعليم الالكتروني نوفر هذه الخدمة لكي نبقيك على اطلاع حول محتوى الملف الذي ستقوم بتحميله .