Absorption laws in Boolean algebra

Question

Does anyone know how to prove the absorption laws in Boolean algebra?

i.e.

$$x + (x * y) = x$$

$$x * (x + y) = x$$

Thankyou so much

Answer 1

If you believe in identity $(x*1 = x)$, the distributive property $(x*(x+y) = x*x + x*y)$, and annulment $(x+1 = 1)$, you may prove it like this:

$$\begin{align} x+(x*y) &&\\ = (x*1)+(x*y) && \text{(identity)}\\ = x*(y+1) && \text{(distributive)}\\ = x * 1 && \text{(annulment)}\\ = x && \text{(identity)}\\ \end{align}$$

The ‘other’ absorption law follows from the fact that

$$x+(x*y) = (x+x) * (x+y) = x * (x+y)$$

Answer 2

Well, as you haven't given any context, there are two layers in axiomatic systems, syntax and semantics. For instance, in lattices, the absorption laws are often part of the axiomatic system. Then if you assign meaning/semantics to the logical formulas, the laws should be tautologies (evident).

Same here, if you have given the variables a range (universe) and assigned meaning to the operators, the laws should be provable to hold. This is called correctness.

Here $x\vee (x\wedge y)$ is evaluated to true if $x$ is true. Same holds for the r.h.s. $x$.

If $x$ is false, $x\vee (x\wedge y)$ is evaluated to false. Same holds for the r.h.s. $x$.