To implement the Boolean function $Y = A'B'C'D + A'BC'D + A'B'CD' + A'BCD + AB'C'D' + AB'CD + ABC'D + ABCD'$ using transmission gates, it's helpful to simplify the function first using Boolean algebra, if possible, and then convert it into a suitable form for implementation with transmission gates.

Step 1: Simplify the Boolean Function

Let's start by grouping terms and see if we can simplify the expression:

$Y = A'B'C'D + A'BC'D + A'B'CD' + A'BCD + AB'C'D' + AB'CD + ABC'D + ABCD'$

Grouping terms:

$Y = A'(B'C'D + BC'D + B'CD' + BCD) + A(B'C'D' + BCD + BC'D + B'C'D')$

Let's simplify each group separately.

1. For $A'$: $B'C'D + BC'D + B'CD' + BCD = C'D(B' + B) + CD'(B' + B) = C'D + CD' = C \oplus D$ So, $A'(C \oplus D)$.

2. For $A$: $B'C'D' + BCD + BC'D + B'C'D' = C'D'(B' + B) + CD(B' + B) = C'D' + CD = C \oplus D$ So, $A(C \oplus D)$.

Thus, the simplified function is:

$Y = A'(C \oplus D) + A(C \oplus D)$

This simplifies further as:

$Y = C \oplus D$

Step 2: Implement the Simplified Function Using Transmission Gates

Now that the Boolean function $Y = C \oplus D$ (which is an XOR function) is simplified, you can implement it using transmission gates.

Transmission Gate XOR Implementation:

1. Inputs: $C$ and $D$.
2. Outputs: $Y = C \oplus D$.

The XOR function can be implemented as follows:

• Output Y = C \oplus D:
  • Use two transmission gates:
    • For $Y = C$ when $D = 0$.
    • For $Y = D$ when $C = 0$.

Implementation steps:

1. Gate 1: Connect input $C$ to output $Y$ and control with $D'$.
2. Gate 2: Connect input $D$ to output $Y$ and control with $C'$.
3. Combine the outputs of the two transmission gates.

Circuit Description:

• Transmission Gate 1: Controlled by $D'$, passes $C$.
• Transmission Gate 2: Controlled by $C'$, passes $D$.

This will produce the output $Y = C \oplus D$, which is the simplified version of your original function.

Verification:

To ensure that the simplified implementation correctly represents the original function, verify each case:

• When $C = 0$ and $D = 0$, $Y = 0$.
• When $C = 0$ and $D = 1$, $Y = 1$.
• When $C = 1$ and $D = 0$, $Y = 1$.
• When $C = 1$ and $D = 1$, $Y = 0$.

Conclusion:

The original Boolean function can be implemented with just an XOR gate using transmission gates, significantly simplifying the circuit.

Would you like more details or have any questions?

Here are some relative questions to consider:

1. What are transmission gates, and how do they function in digital circuits?
2. Can other logic gates be implemented using transmission gates similarly?
3. What are the advantages of using transmission gates for implementing XOR?
4. How would you verify the correctness of the XOR gate implemented with transmission gates?
5. Can this implementation be optimized further in terms of power consumption?

Tip: When simplifying Boolean expressions, always check for common factors and use Boolean algebra to minimize the logic before implementation.