Give step-by-step solution with explanation and final answer:Build a circuit that receives three input bits A, B. C. The circuit should output the “majority”
function. Namely, it should output 1 if t

Question

Give step-by-step solution with explanation and final answer:Build a circuit that receives three input bits A, B. C. The circuit should output the “majority”
function. Namely, it should output 1 if two or more variables among 4, B.C’ are of value 1. Otherwise
the circuit should output 0. Use the following steps in your solution: (a) present a truth table for the
corresponding function, (b) write a corresponding algebraic expression, and (c) draw your circuit.

VIPSolutions · Accepted Answer

## Step-by-Step Solution

### (a) **Truth Table**

Let's list all possible values for inputs $A$, $B$, $C$ and the output:

| A | B | C | Output (Maj) |
|---|---|---|--------------|
|  |  |  |             |
|  |  | 1 |             |
|  | 1 |  |             |
|  | 1 | 1 |      1       |
| 1 |  |  |             |
| 1 |  | 1 |      1       |
| 1 | 1 |  |      1       |
| 1 | 1 | 1 |      1       |

The output is 1 if **at least two inputs are 1**.

---

### (b) **Algebraic Expression**

We need output 1 for cases where at least two variables are 1. Let's write the minterms:

- $A=1, B=1, C=$: $A B \overline{C}$
- $A=1, B=, C=1$: $A \overline{B} C$
- $A=, B=1, C=1$: $\overline{A} B C$
- $A=1, B=1, C=1$: $A B C$

So, the sum of products is:
$$
\text{Maj}(A,B,C) = AB + AC + BC
$$
This covers all cases where at least **two** variables are 1.

---

### (c) **Circuit Diagram**

#### **Required Gates**
- **AND gates** for $AB$, $AC$, $BC$
- **OR gate** to combine them

#### **Diagram**

```
        A -------|&|----\
                 |      |
        B -------      ( )
                 |      |-----|≥1|
        C -------|&|----/     |  |---- Output
                        |-----|≥1|
        A -------|&|----/ 
        |
        C -------

```
Or, more traditionally:

1. **Three 2-input AND gates**:
   - First AND: inputs $A, B$ → output $AB$
   - Second AND: inputs $A, C$ → output $AC$
   - Third AND: inputs $B, C$ → output $BC$
2. **One 3-input OR gate**:
   - Inputs: outputs of all three AND gates ($AB, AC, BC$)
   - Output: $\text{Maj}(A,B,C)$

---

## **Final Answer:**

1. **Truth Table:**  
   See above.

2. **Algebraic Expression:**  
   $$
   \boxed{Maj(A,B,C) = AB + AC + BC}
   $$

3. **Circuit Diagram:**  
   - Three 2-input AND gates (for $AB$, $AC$, $BC$)
   - One 3-input OR gate (to combine them)

**(You may draw the circuit as described above for a complete solution.)**

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