Boolean Algebra A + AB = A

Question

Hi I have a question about the following algebra rule

A + AB = A

My textbook explains this as follows A + AB = A This rule can be proved as such:

Step 1:

Dustributive Law:

A + AB = A*1 = A(1+B) Huh...? Where do they get the one(1) from?

Step 2:

1 + B = 1 {Question:1 + B = B right? so how is this posible(1 + B = 1)?}

Step 3:

A + 1 = A

Thus A + AB = A

If anyone can clarify this for me it would be greatly appreciated

Answer 1

Well, note that $A$ and $AB$ have a common factor--namely $A$--and factoring it out, we have $$A+AB=A(1+B)=A(1)=A$$

Answer 2

$A + AB = (A * 1) + (A * B) = A * (1 + B) = A * 1 = A$.

$A = A * 1$ because $1$ is an identity for $*$, and so $x * 1 = x$ for all $x$. There's not really much more to it.