 |
| basic rules of boolean algebra |
BOOLEAN ALGEBRA'S RULES AND LAWS:
A English mathematician, philosopher and logician George Boole 1854 given a mathematical theories and few logical algebra named as Boolean Algebra..which consist of 12 laws or rule which briefly described Boolean Algebra which are:
1.
A + 0 = A
2.
A + 1 = 1
3.
A . 0 = 0
4.
A . 1 = A
5.
A + A = A
6.
A + Ā = 1
7.
A . A = A
8.
A . Ā= 0
9.
ﬢ(ﬢA) = A
10.
A + A B = A
11.
A + ĀB = A + B
12.
(A +
B)(A + C) = A + BC
Derivation of Laws Using Gates And Truth Table:
RULE # 1 ( A + 0 = A)
 |
| A + 0 = A |
A
|
0
|
A+0
|
0
|
0
|
0
|
1
|
0
|
1
|
RULE # 2 (A + 1 = 1)
 |
| A + 1 = 1 |
A
|
1
|
A+1
|
0
|
1
|
1
|
1
|
1
|
1
|
RULE # 3 ( A . 0 = 0)
 |
| A .0 =0 |
A
|
0
|
A . 0
|
0
|
0
|
0
|
1
|
0
|
0
|
RULE# 4 (A . 1 = A)
 |
| A. 1= A |
A
|
1
|
A . 0
|
0
|
1
|
A
|
1
|
1
|
A
|
RULE #5 ( A + A = A)
 |
| A + A = A. |
A
|
A
|
A +A
|
0
|
0
|
A
|
1
|
1
|
A
|
RULE # 6(A + Ā = 1)
 |
| A + Ā = 1 |
A
|
Ā
|
A+ Ā=1
|
0
|
1
|
1 |
1
|
0
|
1 |
RULE # 7(A . A = A)
 |
| A.A=A |
A
|
A
|
A .A
|
0
|
0 |
A
|
1
|
1 |
A
|
RULE # 8 ( A . Ā= 0)
A . Ā= 0
A
|
Ā
|
A . Ā
|
0
|
1
| 0 |
1
|
0
| 0 |
RULE # 9 (ﬢ(ﬢA) = A)
 |
| A Double not = A |
RULE #10 ( A + A B = A)
= A+ AB
= A (1 + B)
= A (1) (from rule no 2 A + 1= 1, replace the variable A by B so B+1 or 1+B=1)
= A
 |
| A + AB = A |
RULE # 11( A + ĀB = A + B)
= A + ĀB
= A + AB + ĀB (from rule no 10,A+AB=A)
= A + B(A +Ā )
= A + B(1) (from rule no 6, A +Ā = 1)
= A + B (which is equal to right hand side)
 |
| A + ĀB = A + B |
RULE#12(. (A + B)(A + C) = A + BC)
= (A + B)(A + C)
= AA + AC + AB + BC
= A + AC + AB + BC (from rule no 7 A.A = A)
= A(1 + C) + AB + BC
= A (1) + AB +BC (from rule no 2, 1+C=1)
= A + AB + BC
= A + BC (from rule no 10 , A + AB = A) hence it is equal to right hand side
 |
| (A+B)(A+C)=A+AB |
physics label
