# Majority Function: Step-by-Step Solution The **majority function** \( Maj(A, B, C) \) outputs `1` if at least two of its three inputs are `1`. Here’s how to analyze and implement it: --- ## 1. Truth Table | A | B | C | \( Maj(A, B, C) \) | |---|---|---|---------------------| | | | | | | | | 1 | | | | 1 | | | | | 1 | 1 | 1 | | 1 | | | | | 1 | | 1 | 1 | | 1 | 1 | | 1 | | 1 | 1 | 1 | 1 | --- ## 2. Algebraic Expression (Sum of Products) Consider which cases output `1`: - (,1,1) - (1,,1) - (1,1,) - (1,1,1) Each has **at least two 1s**. So, the minimal expression is: \[ \boxed{Maj(A, B, C) = AB + AC + BC} \] - \(AB\): A and B are 1 - \(AC\): A and C are 1 - \(BC\): B and C are 1 --- ## 3. Circuit Diagram (Description) To implement \( Maj(A, B, C) \): - **Three 2-input AND gates:** - One for \( AB \) - One for \( AC \) - One for \( BC \) - **One 3-input OR gate:** - Combine outputs of the three AND gates **Diagram Description:** ``` A ----|&|--\ | |--|≥1|--- Maj(A,B,C) B ----|&|--/ | | A ----|&|------| | | C ----|&|------| | B ----|&|------| | C ----|&|------| ``` *(Replace "|&|" with AND gates and "|≥1|" with a 3-input OR gate. See text for logical connections.)* **Alt text:** *Schematic showing A, B, and C each feeding into three AND gates (AB, AC, BC), whose outputs connect to a 3-input OR gate yielding the majority output.* --- ## 4. Final Answers- **Truth Table:** Already shown above. - **Algebraic Expression:** \[ \boxed{Maj(A, B, C) = AB + AC + BC} \] - **Circuit Diagram:** Three 2-input AND gates for \( AB \), \( AC \), \( BC \) - One 3-input OR gate to combine them --- **Summary:** The majority function for three inputs can be implemented using three -input AND gates (for all input pairs) and a single 3-input OR gate. The algebraic expression is \( Maj(A, B, C) = AB + AC + BC \).