To write a simplified Boolean expression for the given truth table, we’ll analyze each row to see where the output is 1. From your truth table image, we can observe the following:

Truth Table Analysis

The variables are labeled as follows:

• $A$ and $\overline{A}$
• $B$ and $\overline{B}$
• $C$ and $\overline{C}$
• $D$ and $\overline{D}$

The output is 1 in specific rows, and we will identify those combinations and write the corresponding terms.

Reading Each Row with Output 1

From the truth table:

1. Row 1: $\overline{D} \overline{C} \overline{B} \overline{A} = 1$
2. Row 2: $\overline{D} \overline{C} B \overline{A} = 1$
3. Row 3: $D \overline{C} \overline{B} \overline{A} = 1$
4. Row 4: $D \overline{C} B \overline{A} = 1$

Writing the Boolean Expression

Each row with output 1 can be written as a product term (ANDed expression). Combining these with OR (since any of these being true results in the output being 1), we get:

$F = \overline{D} \overline{C} \overline{B} \overline{A} + \overline{D} \overline{C} B \overline{A} + D \overline{C} \overline{B} \overline{A} + D \overline{C} B \overline{A}$

This expression can potentially be simplified further, but that requires analysis of common factors.

Simplification (if possible)

Observe that:

• $\overline{C} \overline{A}$ is a common factor across all terms.

So we can factor out $\overline{C} \overline{A}$:

$F = \overline{C} \overline{A} (\overline{D} \overline{B} + \overline{D} B + D \overline{B} + D B)$

Now, inside the parentheses:

• $\overline{D} \overline{B} + \overline{D} B = \overline{D}$
• $D \overline{B} + D B = D$

Thus, we can further simplify:

$F = \overline{C} \overline{A} (\overline{D} + D)$

Since $(\overline{D} + D) = 1$, the expression simplifies to:

$F = \overline{C} \overline{A}$

Final Simplified Boolean Expression

The simplified Boolean expression for this truth table is:

$F = \overline{C} \overline{A}$

This means the output is 1 whenever $C$ and $A$ are both 0 (hence, $\overline{C}$ and $\overline{A}$ are both 1).

Would you like further details on the simplification process or have any questions?

Further Questions

1. How would this expression change if additional rows were added?
2. What is the significance of finding common factors in Boolean simplification?
3. How does Karnaugh mapping assist in simplifying Boolean expressions?
4. Could this truth table be minimized further if different inputs were used?
5. How do we interpret rows that result in an output of 0?

Tip

In Boolean algebra, always look for common factors to simplify expressions; factoring often reveals opportunities to minimize.