Question

Consider three Boolean variables \( x, y, z \). A majority function outputs 1 if the majority of its inputs are 1, otherwise, it outputs 0. Derive the Boolean expression for the majority function and simplify it using Boolean algebra.

• A. \( xy + yz + xz \)
• B. \( x + y + z \)
• C. \( x(y \oplus z) + yz \)
• D. \( (x \oplus y) \oplus z \)

Hint:

In logic circuits, the majority function is useful in voting systems and fault-tolerant computing, where outputs depend on the majority of inputs.

Answer 1

The Correct Option is A

Understanding the Majority Function.
A majority function for three variables \( x, y, z \) is defined as: \[ f(x,y,z) = 1 \text{ when at least two inputs are 1.} \] The sum of minterms where the function evaluates to 1 is: \[ \Sigma(3,5,6,7) \] Using Boolean algebra, the canonical sum-of-products (SOP) expression for the majority function is: \[ f(x,y,z) = xy + yz + xz \] We can also derive this using simplifications: 1. Expanding in terms of the majority condition: \[ xy + yz + xz \]
2. Alternative simplification using XOR: \[ f(x,y,z) = x(y \oplus z) + yz \] Since both derivations are equivalent, the minimal expression is: \[ xy + yz + xz \]