I know how the distributive law is defined in propositional logic:
$a \vee (b \wedge c) = (a \vee b) \wedge (a \vee c)$
In boolean algebra, conjunction (and) translates to boolean products, while disjunction (or) translates to boolean sums, and my book defines the distributive law as follows:
$a + bc = (a + b)(a + c)$
I'm a bit confused when this law is applied, though. They presented a problem in which one of the steps was:
$y + zy$
And they translated this to:
$y(1 + z)$
Now, in standard algebra, this is pretty straightforward, but it doesn't seem to make sense in terms of the definition they provided. In this case, $y = a = c$ and $z = b$, so by the definition above:
$y + zy = (y + z)(y + y)$
By the idempotent law, this should be:
$y(y + z)$
Could someone please help clarify this? I'm very confused.
They are all equivalent:
$$y(y+z)=yy+yz=y+yz=y(1+z)$$
since
$$yy=y$$