Question

A 3-input majority logic gate has inputs \( X \), \( Y \), and \( Z \). The output \( F \) of the gate is logic ‘1’ if two or more of the inputs are logic ‘1’. The output \( F \) is logic ‘0’ if two or more of the inputs are logic ‘0’.
Which one of the following options is a Boolean expression of the output \( F \)?

• A. \( XY + YZ + ZX \)
• B. \( X \oplus Y \oplus Z \)
• C. \( X + Y + Z \)
• D. \( XYZ \)

Hint:

For majority gates, the output is '1' when at least two inputs are '1'. The expression \( XY + YZ + ZX \) effectively represents this logic.

Answer 1

The Correct Option is A

In a 3-input majority gate, the output \( F \) is '1' when two or more inputs are '1'. Therefore, \( F \) is '1' when at least two of the inputs \( X \), \( Y \), or \( Z \) are '1'. The Boolean expression for this condition is: \[ F = XY + YZ + ZX. \] This expression gives '1' when two or more inputs are '1'. For instance:
\( X = 1, Y = 1, Z = 0 \) gives \( F = 1 \).
\( X = 0, Y = 1, Z = 1 \) gives \( F = 1 \).
\( X = 1, Y = 0, Z = 1 \) gives \( F = 1 \).
Thus, the Boolean expression for the output \( F \) is \( F = XY + YZ + ZX \), which corresponds to option A.