The image provided asks for constructing a truth table for the Boolean function:

$f(x, y) = (x + y') \cdot (xy + y')'$

Let's break down the steps required to fill in the truth table and evaluate $f(x, y)$.

1. Column for $x$ and $y$: These are given as standard truth values for all possible combinations (0, 0), (0, 1), (1, 0), and (1, 1).

2. $y'$: This is the complement of $y$.

   • When $y = 0$, $y' = 1$.
   • When $y = 1$, $y' = 0$.

3. $x + y'$: This is the logical OR of $x$ and $y'$.

4. $xy$: This is the logical AND of $x$ and $y$.

5. $xy + y'$: Logical OR of the previous results (AND and $y'$).

6. $(xy + y')'$: The complement of the OR operation in the previous step.

7. $f(x, y) = (x + y') \cdot (xy + y')'$: The final AND operation between the two main terms.

Let's now complete the table:

So, the final values of $f(x, y)$ for all input combinations are $0$ in every case.

Would you like more details or clarification on any of these steps?

Here are 5 related questions:

1. What is the meaning of the complement operator $'$ in Boolean algebra?
2. How do logical AND and OR operations work in Boolean expressions?
3. Can this Boolean expression $f(x, y)$ be simplified further?
4. What would happen if we added a third variable, like $z$, to the function?
5. How can Boolean functions be represented using logic gates?

Tip: Boolean algebra is crucial in simplifying complex logical expressions, which can help in designing efficient digital circuits.