The eminent English mathematician George Bull first discussed Boolean algebra in his first book, The Mathematical Analysis of Logic, published in 1847. It discusses based on truth and falsehood. Even after the invention of the binary number system, it is possible to solve all the mathematical problems of arithmetic in the computer by changing the truth and falsehood of algebra with binary 1 and 0.
In general algebra, there can be different values of variables. But in Bolivian algebra, only two values of a variable can be true (1) or false (0).
0 (0 volt to 0.8 volt)
1 (2 volt to 5 volt)
The electronic circuits of the computer do not contain any value between 0 and 1.
★ Boolean algebra has three basic functions: -
1.
Boolean addition operation (OR operation)
2. Boolean multiplication action (AND operation)
3.
Boolean complementary action (NOT operation)
★ No use of algebraic fractions, negative numbers, squares, etc.
1.
The sign denotes the OR.
2.
AND is denoted by ⋀.
3. NOT is denoted by the 〜 sign.
Dual policy-
All the Boolean algebra theorems used to determine a valid equation from one valid equation to another are called Boolean duality. In Boolean algebra, all the theorems related to OR and AND follow a dual principle. All theorems related to AND and OR operations follow the dual rule.
AND (.) And OR (+) exchange. Such as: -
1 + 1 = 1
1. 1 = 1
1 and 0 are exchanged,
0 + 1 = 1
1 + 0 = 1
And
1. 0 = 0
0. 1 = 0
Boolean theorems:
The theorems used by George Bull to provide mathematical representations of all kinds of rational subjects are called Boolean theorems. The rules by which Boolean algebras are solved are called Boolean theorems. Boolean theorems can be used to simplify logic numbers or change the structure of numbers.
These theorems can be easily proved with a value of Boolean variable 1 or 0. Below are some important Boolean theorems.
Below are some important Boolean theorems
The basic theorem
1. i) A +1 = 1 ii) A.1 = A
2. i) A + 0 = A ii) A.0 = 0
3. i) A + A = 1 ii) A.A = 0
4. i) A + A = A ii) A.A = A
Auxiliary Theorems
i) X+ XY= X+ Y
ii) X+ XY= X+Y
iii) X+ XY= X+ Y
iv) X+ XY= X+ Y
Commutative Theorems
i) X+ Y= Y+ X
ii) X.Y = Y.X
Associative Theorems
i) M + (N + O) = (M+ N) + O
ii) M. (N.O) = (M.N) .O
Distributed Theorems
1.X (Y+ Z) = XY + XZ
2.(P + Q) (P + R) = P+ QR
De-Morgan’s Theorems
i) A + B = A.B ii) A.B = A + B
Secondary Theorems
i) A + AB = A ii) A (A + B) = A
11. A = A
The above theorems can be proved by any theorem with a value of variable 0 or 1.
