Boolean Logic Equation

Question

How can I prove this. what is the way? im up to the second last line. and i dont actually know how can x * (1 + y) then the y just disappears into x * 1.

x + x * y = x
x + x * y = (x * 1) + (x * y)
          = x * (1 + y)
          = x * 1 [THIS LINE]
          = x

thanks

Answer 1

In Boolean Algebra we have:

\begin{align*} a \cdot b &= \min{(a,b)}\\ a + b &= \max{(a,b)} \end{align*}

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So, by the first definition it's obvious that:

$$ x \cdot 1 = \min{(x,1)} = x$$

Since $x \leq 1$.

By the same reasoning:

$$1 + y = \max{(1,y)} = 1$$ since $y \leq 1$.

So you have:

\begin{align*} x + x \cdot y &= (x \cdot 1) + (x \cdot y)\\ &= x \cdot \color{red}{(1 + y)}\\ &= x \cdot \color{red}{1}\\ &= x \cdot 1 = x \end{align*}