Absorption Law Proof by Algebra

I'm struggling to understand the absorption law proof and I hope maybe you could help me out.

The absorption law states that: $X + XY = X$
Which is equivalent to $(X \cdot 1) + (XY) = X$

No problem yet, it's this next step that stumps me. How can I apply the distributive law when there are two "brackets"?

How can I manipulate $(X \cdot 1) + (XY) = X$ to give me $X \cdot (1+Y)$?

I understand that the absorption law works. I would just like to see how the algebra proof works.

Thank you!

3 Answers

First of all, this is a math question.

I think you are confused on how brackets are used. They indicate precedence of operations, and can be used anywhere, even in places where such indication is not necessary. For example, $$3 \times 5 + 8$$ and $$(3 \times 5) + 8$$ are both legitimate expressions and they mean exactly the same thing.

Your expression $$(x \cdot 1) + (x \cdot y)$$ is exactly the same as $$x \cdot 1 + x \cdot y.$$ So we may calculate $$(x \cdot 1) + (x \cdot y) = x \cdot (1 + y) = x \cdot 1 = x.$$

Apply the distributive law on $$x\land(1\lor y)$$ and see what you get.

Try and think about it the other way. Can you legally go from the step below to the step above? If yes then it is fine( at least in this case) to go from top to bottom too.