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The image shows a digital circuit problem that asks to derive the function of a logic circuit involving 2-to-4 decoders and AND gates.

Here's a step-by-step approach to solve each part of the problem:

### Step 1: Understanding the Circuit
- **Inputs:** There are four inputs to the system: \(A\), \(B\), \(C\), and \(D\).
- **Decoders:** There are two 2-to-4 decoders. Each decoder takes two inputs:
  - One decoder takes \(A\) and \(B\) as inputs and produces outputs labeled "1-HOT".
  - The second decoder takes \(C\) and \(D\) as inputs and produces outputs labeled "0-COLD".
- **AND Gates:** Each decoder's output goes into a corresponding AND gate, and the output of the AND gates feeds into a final OR gate producing \(F\).

### a. Deriving the 4-variable truth table
To derive the truth table, we need to evaluate all combinations of \(A\), \(B\), \(C\), and \(D\) (16 possible combinations) and how the decoders and gates process these inputs.

A 2-to-4 decoder with inputs \(A\) and \(B\) produces four outputs:
- Decoder 1 (1-HOT): \( \overline{A} \overline{B}, \overline{A} B, A \overline{B}, A B \)
- Decoder 2 (0-COLD): \( \overline{C} \overline{D}, \overline{C} D, C \overline{D}, C D \)

Each AND gate combines one output from the first decoder with one from the second. The final OR gate outputs the sum of these products.

We'll calculate the output for each combination of inputs.

### b. Deriving the K-map
Using the truth table from part (a), we can create a Karnaugh map (K-map) to simplify the Boolean expression.

### c. Deriving the minimal SOP and POS
From the K-map, the minimal Sum of Products (SOP) and Product of Sums (POS) forms can be derived, which are the simplified Boolean expressions representing the circuit.

### d. Designing the circuit using a 4:1 multiplexer and logic gates
For this part, we need to design a circuit equivalent to the given one, but using a 4:1 multiplexer instead of decoders and AND gates. This would involve mapping the inputs \(A\) and \(B\) as the select lines and using appropriate logic for the multiplexer inputs.

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Would you like to proceed with the detailed truth table and K-map explanation for part (a) and part (b)?

Here are 5 related questions that expand on this topic:
1. How do 2-to-4 decoders function in digital logic circuits?
2. What is the purpose of using a Karnaugh map in simplifying Boolean expressions?
3. How can the minimal SOP form be implemented using basic logic gates?
4. What are the advantages of using a 4:1 multiplexer in circuit design?
5. How does a product of sums (POS) expression differ from a sum of products (SOP)?

**Tip:** When working with Karnaugh maps, grouping as many adjacent ones or zeros as possible helps in minimizing the Boolean expression effectively.

Refer to the circuit diagram to solve the following:
 a) Derive the 4-variable truth table
 b) Derive the K-map
 c) Derive the minimal SOP and POS
 d) Design a circuit that represents F using a 4:1 multiplexor and logic gates.

Analyze a digital logic circuit using decoders, AND gates, and an OR gate to derive a truth table, K-map, minimal SOP and POS expressions, and design the circuit using a 4:1 multiplexer. This problem is suitable for undergraduate electronics or digital design students.

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