Prove following rules by the method of perfect induction:
(A+B) .(. )
= A. + .
Answers
Answer:
Step-by-step explanation:
There are at least two paths to demonstrate a theorem: the classic algebraic method and perfect induction case, very useful in Boolean Algebra.
This last path says that if you check the veracity of a theorem for all possible input combinations, then the theorem is true in its entirety. This is, if it is fulfilled in each case, it is fulfilled in general. This path can be used in Boolean Algebra since the variables have only two possible values: 0 and 1, whilst in our algebra each variable can have infinite values.
5. X . 0 =0
6. X . 1 =X
7. X . X =X
__
8. X . X =0
==
9. X = X
Boolean Algebra
16. X + XZ = X
17. X(X+Z)=X
__
18. X+ X Y =X+Y
__
19. X ( X +Y) =X.Y
____ __ __
20. X+Y = X . Y
____ __ __
21. X.Y =X+Y
Laws of absorption Identity Theorems
De Morgan's Theorems
262
AND operations
Double complement
10. X + Y = Y + X
11. XY=YX
12. (X + Y ) +Z = X +(Y + Z) Associative laws
13. (X . Y). Z =X. (Y. Z)
14. X (Y + Z) = XY + XZ Distribution Law
15. X + Y .Z = (X + Y) . (X + Z) Dual of Distributive Law
Commutative laws
Proof of Boolean Algebra Rules:
Every rule can be proved by the application of rules and by perfect Induction.
Rule 15:
(i) This rule does not apply to normal algebra We follow:
(X + Y) (X + Z) = XX + XZ +YX + YZ
=X+ XZ +YX + YZ, X.X=X