Understanding the Boolean Expression

We are given the Boolean expression $A(x, y, z) = x (y' z)'$. The goal is to find its complete sum-of-products (SOP) form. A complete SOP means that every product term in the expression must contain all the input variables (x, y, and z), either in their original or complemented form.

Simplifying the Expression

First, let's simplify the given expression using Boolean algebra rules:

1. Apply De Morgan's Law to the term $(y' z)'$. De Morgan's Law states that $(P Q)' = P' + Q'$. Applying this, we get: $(y' z)' = (y')' + z'$
2. Simplify $(y')'$ using the involution law, which states $(P')' = P$: $(y')' + z' = y + z'$
3. Substitute this back into the original expression: $A(x, y, z) = x (y + z')$
4. Apply the distributive law: $P(Q + R) = PQ + PR$: $A(x, y, z) = x y + x z'$

Converting to Complete Sum-of-Products (SOP)

The simplified expression is $xy + xz'$. This is a sum of products, but it is not yet a *complete* SOP because the terms $xy$ and $xz'$ do not contain all three variables (x, y, z).

We need to expand each term so that it includes all variables:

1. Expand the term $xy$:

   This term is missing the variable $z$. We can introduce $z$ and its complement $z'$ using the identity $P = P(Q + Q') = PQ + PQ'$.

   $xy = xy (z + z')$ $xy = xyz + xy z'$
2. Expand the term $xz'$:

   This term is missing the variable $y$. We introduce $y$ and its complement $y'$ using the same identity:

   $xz' = xz'(y + y')$ $xz' = xz'y + xz'y'$

   Reordering the variables alphabetically within each term:

   $xz' = xyz' + xy'z'$
3. Combine the expanded terms:

   Now, substitute the expanded terms back into the expression:

   $A(x, y, z) = (xyz + xyz') + (xyz' + xy'z')$
4. Simplify by removing duplicate terms:

   The term $xyz'$ appears twice. In Boolean algebra, $P + P = P$. So, we remove the duplicate:

   $A(x, y, z) = xyz + xyz' + xy'z'$

This expression $xyz + xyz' + xy'z'$ is now in the complete sum-of-products form, as each product term contains all three variables x, y, and z.

Comparing with Options

Let's compare our result $xyz + xyz' + xy'z'$ with the given options:

• Option 1: $xyz + x'yz' + x'y'z'$ - Incorrect.
• Option 2: $xyz' + x'yz' + x'y'z'$ - Incorrect.
• Option 3: $xyz" + xy'z + x'y'z'$ (Assuming " is a typo for ') - Incorrect.
• Option 4: $xyz + xyz' + xy'z'$ - Correct.

Our derived complete SOP matches Option 4.