Basic Theorem and properties of Boolean Algebra

1). A+0=A

2). A+1=1

3). A.0=0

4). A.1=A

5). A+A=A

6). A+A’=1

7). A.A=A

8). A.A’=0

9). A”=A

10). A+AB=A

A + AB = A(1 + B)    ∴ Factoring (distributive law)
A + AB = A.1         ∴ Rule 2: (1 + B)= 1
A + AB = A           ∴ Rule 4: A .1 = A

11). A+A’B=A+B

=(A+A') (A+B)        ∴ x+yz=(x+y)(x+z)
=1(A+B)
=A+B

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

= AA + AC + AB + BC          ∴ Distributive law
= A + AC + AB + BC           ∴ Rule 7: AA = A
= A( 1 + C)+ AB + BC         ∴ Rule 2: 1 + C = 1
= A.1 + AB + BC              ∴ Factoring (distributive law)
= A(1 + B)+ BC               ∴ Rule 2: 1 + B = 1
= A.1 + BC                   ∴ Rule 4: A .1 = A
= A + BC