- #1

mathmari

Gold Member

MHB

- 5,049

- 7

We have the following equalities:

\begin{align*}\left (\overline{y}\land z\lor x\land \left (\overline{z}\lor y\right )\right )\land \left (\overline{x}\lor \overline{y}\right )& \overset{(1)}{=}\left (\overline{y}\land z\lor x\land \overline{z}\lor x\land y\right )\land \left (\overline{x}\lor \overline{y}\right ) \\ & \overset{(2)}{=}\left (\overline{y}\land z\lor x\land \overline{z}\right )\land \left (\overline{x}\lor \overline{y}\right ) \\ & \overset{(3)}{=}\left (\overline{y}\land \left (z\lor x\land \overline{z}\right )\lor y\land x\land \overline{z}\right )\land \left (\overline{x}\lor \overline{y}\right ) \\ & \overset{(4)}{=}\overline{y}\land \left (x\lor z\right )\land \left (\overline{x}\lor \overline{y}\right )\lor y\land x\land \overline{z}\land \left (\overline{x}\lor \overline{y}\right )\\ & \overset{(5)}{=}\overline{y}\land \left (x\lor z\right ) \lor \overline{\overline{y}\lor \overline{x}}\land \overline{z}\land \left (\overline{x}\lor \overline{y}\right ) \\ & \overset{(6)}{=}\overline{y}\land \left (x\lor z\right )\end{align*}

I want to justify at each step which transformations have been done. First of all, $\left (\overline{y}\land z\lor x\land \left (\overline{z}\lor y\right )\right )\land \left (\overline{x}\lor \overline{y}\right )$ is equivalent to $\left (\overline{y}\land z \lor \left [x\land \left (\overline{z}\lor y\right )\right ]\right )\land \left (\overline{x}\lor \overline{y}\right )$, or not? Then applying the distributive law we get $\left (\overline{y}\land z \lor \left [\left (x\land \overline{z}\right )\lor \left (x\land y\right )\right ]\right )\land \left (\overline{x}\lor \overline{y}\right )$. ow we can eliminat the parentheses and we get $\left (\overline{y}\land z\lor x\land \overline{z}\lor x\land y\right )\land \left (\overline{x}\lor \overline{y}\right ) $ and so the first step is justified.

Is everything correct so far? (Wondering)

How do we continue? How do we get to step $(2)$, i.e. why can we eliminate the term $x\land y$ ? (Wondering)