Draw a circuit to implement 𝑌 = 𝐴𝐵 + 𝐴𝐵𝐶 + 𝐴̅ 𝐵)𝐶̅ using a decoder. Label thedecoder pins appropriately

To implement the given Boolean expression Y = AB + ABC + \overline{A}B)C̅ using a decoder, we will follow these steps:

1. Break Down the Problem

1. The expression can be simplified if needed and can be implemented using a 3-to-8 decoder.
2. We'll determine the inputs and how they relate to the output.

2. Relevant Concepts

• A 3-to-8 decoder takes 3 input signals and produces 8 output signals corresponding to the binary values of the input.
• The output pins will be active high for the corresponding binary encoded input.

3. Analysis and Detail

1. Identify the variables:

   • Inputs: $A, B, C$
   • Outputs: There will be 8 outputs corresponding to combinations of $A, B, C$ (000 to 111).

2. The relevant minterms from the expression $Y$:

   • $AB$ corresponds to minterms $110$ (where $A=1, B=1, C=0$) and $111$ (where $A=1, B=1, C=1$).
   • $ABC$ is already part of $AB$ when $C$ is also true, hence does not introduce additional outputs.
   • $\overline{A}B$ corresponds to $B$ when $A=0$, which describes minterms $010$ (where $A=0, B=1, C=0$) and $011$ (where $A=0, B=1, C=1$).
   • The expression C̅ indicates that we will need outputs from the decoder when $C=0$.

3. Based on these minterms, the active outputs from the decoder will be:

   • Output $2$ (for minterm $010$)
   • Output $3$ (for minterm $011$)
   • Output $6$ (for minterm $110$)
   • Output $7$ (for minterm $111$)

4. Verify and Summarize

The circuit will use outputs $D2, D3, D6, D7$ of the decoder to derive $Y$ by connecting them to an OR gate.

Final Answer

Here's a sample circuit representation:

Decoder (3-to-8)
Inputs: A (pin 1), B (pin 2), C (pin 3)
Outputs: D0 (000), D1 (001), D2 (010), D3 (011), D4 (100), D5 (101), D6 (110), D7 (111)

Connections:
- D2 (from A=0, B=1, C=0)
- D3 (from A=0, B=1, C=1)
- D6 (from A=1, B=1, C=0)
- D7 (from A=1, B=1, C=1)

Final Output:
Y = D2 + D3 + D6 + D7

Circuit Diagram

  A ------|   |                      Y
  B ------|   |--------------------|   |
  C ------|   |        OR         |   |
          | 3-to-8 Decoder       |   |
          |   |                    ---
          |   |                    |
          ---------------------------  

This circuit will successfully implement the function $Y = AB + ABC + \overline{A}BC$ using a decoder.