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# LD CH2: BOOLEAN ALGEBRA

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## Ertugrul Eris

on 9 November 2018

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#### Transcript of LD CH2: BOOLEAN ALGEBRA

BOOLEAN ALGEBRA
BASIC CONCEPTS:RELATION
BOOLEAN ALGEBRA:AXIOMS
BOOLEAN ALGEBRA: THEOREMS
Theorem: Complement element is unique.

Theorem: a+1 = 1 and a. 0 = 0.

Theorem:0'=1 and 1'=0 .

Theorem:Idempotent (eşkuvvet) theorem
a+a = a and a.a = a.
BOOLEAN FUNCTIONS AND REPESENTATION
Cartesian product
of two sets X and Y  denoted X × Y, is the set of all possible ordered pairs whose first component is a member of X and whose second component is a member of Y.
Relation (Bağıntı):
Assume that A «domain» (tanım kümesi), and B «range» (değer kümesi) are two sets, then R «relation»set is
Completely specified relation (Tamamen belirlenmiş bağıntı ) :
If the R relation includes all the elements in the domain of A then it is called «completely specified» relation
Functional relation (Fonksiyonel bağıntı):
In a «R» relation, for each element in A if there exists one and only one element corresponding in B then it is called as «functional relation»
Function:
Completely specified functional relation is called function.
It is shown as F : A→ B .
BASIC CONCEPTS:FUNCTION
EQUIVALENCE RELATION:
If R:A →A has the following properties then it is called as «equivalence relation» and shown as " = ".

Reflexive:

Symmetric:

Transitive:

ORDER RELATION:
If R:A →A has the following properties then it is called as «order relation» and shown as "<".

Symmetric:

Transitive:
Physical:time,
Mathematical model Real numbers (Order relation set)
Which arithmetic operation is an equivalence relation?
n-ary operation/n-variable function(n-lik işlem):
Assume that A^n =AXA...A is domain set and A is range set then
is called n-ary operation or n-variable function.
one-variable function examples: 5cos 2t ; 2x+3
two- variable function examples: x+5y; xy
Boolean Algebra is a mathematical system composed of three elements M = (S,R,A) such as
S: is a
finite-element-set
.
R: is a set of three operations defined on S:
Two 2-ary operations called ‘Summation, OR' and 'product, AND'
One 1-ary operation called ‘complement'.
A: is a set of axioms which defines R operations on S.
Assuming that a,b,c are the elements of S:
Axiom.1 Commutative (Yer değiştirme) axiom
1. a+ b = b+ a
2. a. b = b. a
Axiom.2. distributive (Dağılma) axiom
1. a +(b c) = (a+ b) (a+ c)
2. a. (b+ c) = (a b)+ (a c)
Axiom.3.Unit elements existence
1. a+ 0 = 0+ a = a
2. a .1 = 1 .a = a
Axiom.4.Complement (Tümleyen) element existence
1. a+ a' = a‘+ a = 1
2. a. a' = a' .a = 0
What is (are) the difference(s) between the known algebra and the Boolean Algebra?
DUALITY PRINCIPLE
For each axiom there is a dual axiom which is found by replacing (+) with (.) and (1) with (0)
and vice versa
Conclusion
Once a theorem is proved then it’s dual is also true without any proof.
Theorem:An elements complement’s
Complement is equal to itself (a' )'=a.

Theorem:absorb (Yutma)
a+a.b = a and a.(a+b) = a

Theorem
a+a' b = a+b and a.(a'+b) = ab

Theorem of ‘Associativity’ (guplandırma)
a+(b+c) = (a+b)+c and a.(b.c) = (a.b) .c dir.
De Morgan theorem:
(a+b)' = a' b' and ab)' = a'+b'

Consensus theorem
a.b + a' c + b.c = a.b + a'.c
BOOLEAN ALGEBRA EXAMPLE
Assuming that a,b,c are the elements of S:
Axiom.1 Commutative (Yer değiştirme) axiom
1. a+ b = b+ a
2. a. b = b. a
Axiom.2. distributive (Dağılma) axiom
1. a +(b c) = (a+ b) (a+ c)
2. a. (b+ c) = (a b)+ (a c)
Axiom.3.Unit elements existence
1. a+ 0 = 0+ a = a
2. a .1 = 1 .a = a
Axiom.4.Complement element existence
1. a+ a' = a‘+ a = 1
2. a. a' = a' .a = 0
NUMBER OF FUNCTIONS DEFINED ON A FINETE SET
Repeated N elements' n-tuple permutations is equal to
EXAMPLE
n-variable Boolean Function:

By applying (+), (.), (‘) operations to the x1, x2 ,...xn variables, each function defined is called «Boolean Function».
In other words each function written as a formula is a «Boolean Function»
All the functions other than Boolean are called Non-Boolean.
Fundamental theorem of Boolean Algebra is
Shannon theorem
Generalized Shannon Theorem
BOOLEAN /
NON BOOLEAN FUNCTIONS
"Second type canonical" or
"product of sums terms"
"First type canonical" or
"sum of products terms"
product terms:minterms
sum terms:maxterms
FUNCTION TRUTH TABLE / CANONICAL FORMS EXAMPLES
TRUE MINTERM/TRUE MAXTERM REPRESENTATION EXAMPLE
CANONICAL FORM REPRESENTATION EXAMPLE
BOOLEAN FUNCTIONS
BOOLEAN ALGEBRA
How many different representations are there for Boolean functions?
Assume that
EXAMPLE
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