Axioms and Laws of Boolean Algebra
Boolean algebra devised in 1864 by George Boole, is a system of mathematical logic. It is an algebraic system consisting of a set of element (0,1) associated with a Boolean variable and two binary operators AND and OR and a uniry operator NOT. In mathematics, an identity is a statement true for all possible values of its variable or variables.
2. OR Operation
3. NOT Operation
0’ = 1
1’ = 0
In mathematics, an identity is a statement true for all possible values of its variable or variables.
|Additive Identity||A + 0 = A|
|A + 1 = 1|
|A + A = A|
|A + A’ = 1|
|Multiplicative Identity||A . 0 = 0|
|A . 1 = A|
|A . A = A|
|A . A’ = 0|
(A’)’ = A
0′ = 1 ; If A = 0 then A’ = 1
1′ = 0 ; If A = 1 then A’ = 0
Another type of mathematical identity, called a “property” or a “law,” describes how differing variables relate to each other in a system of numbers
we can reverse the order of variables that are either added together or multiplied together without changing the truth of the expression:
A+B = B+A
This property tells us we can associate groups of added or multiplied variables together with parentheses without altering the truth of the equations.
A + ( B + C ) = ( A + B ) + C = ( A + C ) + B
A(BC) = (AB)C = (AC)B
A(B+C) = AB + AC
A + AB = A
A + AB
A (1 + B)
A.1 ( Applying A + 1 = 1)
Rule-2: A + A’B = A + B
A + A’ B
= A+AB + A’B (Applying A+ AB = A)
= A + B ( A + A’) (Applying A + A’ = 1)
= A + B
Rule-3 for simplifying POS
(A+B)(A+C) = A + BC
=AA + AC + BA + BC ; Applying distributive law
= (A + AC) + AB + BC ; Applying A+ AB=A
= (A + AB) + BC
= A + BC
To summarize, here are the three new rules of Boolean simplification expounded in this section:
A + AB = A
A + A’B = A+ B
(A+B)(A + C) = A + BC
AB + A’C + BC
AB + A’C +BC(A+A’)
AB + A’C + ABC + A’BC
AB + ABC + A’C + A’BC
AB(1+C) + A’C (1+B) ; Using identity Law A+1=1
AB + A’C
AB + A’C + BC = AB + A’C Prooved
(AA’ +AC +A’B + BC)(B+C)
ABC +A’BB + BBC + ACC + A’BC + BCC
ABC +A’B + BC + AC + A’BC + BC
BC(A+1) + A’B(1+C) +AC + BC
BC + A’B + AC + BC
(BC + BC) + A’B + AC
AC + A’B + BC
(A+B)(A’+C)(B+C) = AC + A’B + BC Proved
Theorem AB + A’C = (A+C) (A’+B)
AA’ + AB + A’C + AB
0 + A’C + AB + BC
A’C + AB + BC(A+A’)
A’C + AB + ABC + A’BC
AB(1+C) + A’C(1+B)
AB + A’C Prooved
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12. Which is the Boolean expression from the truth table shown below?
13. When the Boolean function F(A,B,C) = Σm(0,1,2,3)+(4,5,6,7) is minimised. What does one get?
14. Which of the following statement is not correct?
15. A,B,C are three boolaen variables. Which of the following Boolean expressions cannot be minimised further?
16. The minimised form of logical expression (A’B’C’ + A’BC’ +A’BC +ABC’) is ..