Boolean Algebra rules

Boolean Algebra invention started by George Boole. He invented a rule that there is a clear connection between mathematics and logic.   Pre...

Boolean Algebra invention started by George Boole. He invented a rule that there is a clear connection between mathematics and logic. 

Previously I have discussed De Morgan's Law.

Boolean Algebra rules

Boolean Algebra rules have been made by depending on true false. But when the use of binary number system(0,1) started on computers from that time this true-false is represented by binary numbers 1 and 0.

Only for addition and multiplication binary digit 0 and 1 are used in Boolean Algebra. No geometrical, trigonometric rule is not usable in Boolean algebra. Here no negative or fractional number is not usable.

For different simplifications, we need to apply many rules. By applying these rules we can make the equations simple.

Variable 

The value of the Boolean variable is changeable. It changes for time. If A is a Boolean variable then it can take 0 or 1 any of these.

Constant 

In Boolean algebra, 2 digits (0 and 1) are used. Value of 0 and 1 never changes. That is why 0 and 1 are called Boolean constant is a Boolean algebra.

Substitute

Two value of boolean algebra is 0 and 1. These two are the substitute for each other. It is represented by the ' ¯ ' sign. Sometimes ' / ' sign is also used. 

Substitution rules

1.  ͞0 =1
2. 1 = 0
3. Ā̄ = A
4. If A = 0 then Ā = 1 
5. If A = 1 then Ā=0

Basic operations

3 types of Boolean basic operations: 

Logical addition 

It is also called OR operation. "+" sign is usually used in this case. Example: A OR B is represented as A+B.

Logical multiplication

It is also called AND operation. " • " sign is usually used in this case. Example: A AND B is represented as A•B.

Logical inversion

It is complementation or NOT operation. " / " or " ˊ " sign is used in this case. Example: NOT (A) is represented as Aˊ.

Logical operator

In Boolean algebra for completing 3 Boolean operations, we use some signs. Those are called logical operators.

AND operator: The logical multiplication process is done by this.

OR operator: The logical addition process is done by this.

NOT operator: Logical inversion is done by this.

Postulates

All the operations in Boolean algebra are completed by addition and multiplication. In the addition and multiplication process, Boolean algebra follows some rules.

Addition rules

1. 0+0 = 0
2. 0+1 = 1
3. 1+0 = 1
4. 1+1 = 1

Multiplication rules

1. 0.0 = 0
2. 0.1 = 0
3. 1.0 = 0
4. 1.1 = 1

Fundamental

(A) A.0 =0

A	0	A.0
0	0	0
1	0	0

(B) A.1 = A

A	1	A.1
0	1	0
1	1	1

(C) A.A = A

A	A	A.A
0	0	0
1	1	1

(D)  A.Ā = 0

A	Ā	A.Ā
0	1	0
1	0	0

(E) A + 0 = A

A	0	A+0
0	0	0
1	0	1

(F) A + 1 = 1

A	1	A+1
0	1	1
1	1	1

(G) A+A = A

A	A	A+A
0	0	0
1	1	1

(G) A+Ā = 1

A	Ā	A+Ā
0	1	1
1	0	1

 Communicative

A) A+B=B+A

A	B	A+B	B+A
0	0	0	0
0	1	1	1
1	0	1	1
1	1	1	1

B) A.B = B.A

A	B	A.B	B.A
0	0	0	0
0	1	0	0
1	0	0	0
1	1	1	1

 Associative  

(A) A+(B+C) = (A+B) + C

A	B	C	B+C	A+B	A + (B+C)	(A+B)+C
0	0	0	0	0	0	0
0	0	1	1	0	1	1
0	1	0	1	1	1	1
0	1	0	1	1	1	1
1	0	1	1	1	1	1
1	0	0	0	1	1	1
1	1	1	1	1	1	1
1	1	0	1	1	1	1

(B) A(BC) = (AB)C

A	B	C	BC	AB	A (BC)	(AB) C
0	0	0	0	0	0	0
0	0	1	0	0	0	0
0	1	0	0	0	0	0
0	1	0	0	0	0	0
1	0	1	0	0	0	0
1	0	0	0	0	0	0
1	1	1	1	1	1	1
1	1	0	0	1	0	0

Distributive

(A) A(B+C) = AB + AC

A	B	C	AB	AC	B+C	A (B+C)	AB+AC
0	0	0	0	0	0	0	0
0	0	1	0	0	1	0	0
0	1	0	0	0	1	0	0
0	1	0	0	0	1	0	0
1	0	1	0	1	1	1	1
1	0	0	0	0	0	0	0
1	1	1	1	1	1	1	1
1	1	0	1	0	1	1	1

(B) A+BC = (A+B)(A+C)

A	B	C	BC	A+B	A+C	A+BC	(A+B)(A+C)
0	0	0	0	0	0	0	0
0	0	1	0	1	1	0	0
0	1	0	0	1	0	0	0
0	1	0	0	1	0	0	0
1	0	1	0	1	1	1	1
1	0	0	0	1	1	1	1
1	1	1	1	1	1	1	1
1	1	0	0	1	1	1	1

 Some extra laws 

(A) A+AB = A

A	B	AB	A + AB
0	0	0	0
0	1	0	0
1	0	0	1
1	1	1	1

(B) Ā̄ = A

A	Ā	Ā̄
0	1	0
1	0	1

(C) A(A+B) = A

A	B	A+B	A (A+B)
0	0	0	0
0	1	1	0
1	0	1	1
1	1	1	1

(D) A + ĀB = A+B

A	B	Ā	ĀB	A+ĀB	A+B
0	0	1	0	0	0
0	1	1	1	1	1
1	0	0	0	1	1
1	1	0	0	1	1

(E) A(Ā+B) = AB 

A	B	Ā	Ā+B	A(Ā+B)	AB
0	0	1	1	0	0
0	1	1	1	0	0
1	0	0	0	0	0
1	1	0	1	1	1