The minimum number of gates required to realize this expression `Z =DABC + DA bar(BC )` is

Options

A. One
B. Two
C. eight
D. five

Answer

To solve the expression \( Z = DABC + DA'BC \) and determine the minimum number of gates required to realize it, we can follow these steps:

### Step 1: Identify the given expression
The expression given is:
\[ Z = DABC + DA'BC \]

### Step 2: Factor out common terms
We can see that both terms in the expression share a common factor of \( DA \). Thus, we can factor \( DA \) out:
\[ Z = DA(BC + A'BC) \]

### Step 3: Simplify the expression inside the parentheses
Now, we can simplify the expression \( BC + A'BC \). We can factor out \( BC \):
\[ Z = DA(BC(1 + A')) \]
Using the property \( 1 + A' = 1 \):
\[ Z = DA(BC \cdot 1) = DABC \]

### Step 4: Apply Boolean algebra rules
Now we can simplify further. Since \( BC + A'BC \) simplifies to \( BC \) when \( A \) is true, we can rewrite the expression:
\[ Z = DABC + DA'BC = DA(B + A'B) \]
Using the absorption law \( B + A'B = B(1 + A') = B \):
\[ Z = DAB \]

### Step 5: Determine the number of gates required
The final simplified expression is:
\[ Z = DAB \]
This expression can be realized using a single AND gate that takes \( D \), \( A \), and \( B \) as inputs.

### Conclusion
Thus, the minimum number of gates required to realize the expression \( Z = DABC + DA'BC \) is **one AND gate**.