Question

A Boolean function is given as \[ f = (\bar{u} + \bar{v} + \bar{w} + \bar{x}) \cdot (\bar{u} + \bar{v} + \bar{w} + x) \cdot (\bar{u} + v + \bar{w} + \bar{x}) \cdot (\bar{u} + v + \bar{w} + x) \] The simplified form of this function is represented by:

• A.
• B.
• C.
• D.

Hint:

To simplify Boolean expressions, look for common literals across all product terms. If a variable appears complemented in every term, it can be factored out of the expression directly.

Answer 1

The Correct Option is A

Let's simplify the Boolean expression step by step. All four terms in the expression contain \( \bar{u} \) and \( \bar{w} \), which means: \[ f = \bar{u} \cdot \bar{w} \cdot (\text{some other terms}) \] From the expression: \[ \begin{aligned} f &= (\bar{u} + \bar{v} + \bar{w} + \bar{x})(\bar{u} + \bar{v} + \bar{w} + x) \\ &\quad \cdot (\bar{u} + v + \bar{w} + \bar{x})(\bar{u} + v + \bar{w} + x) \end{aligned} \] Factor out \( \bar{u} \) and \( \bar{w} \) from all terms: \[ f = \bar{u} \cdot \bar{w} \] Therefore, the simplified expression is: \[ f = \bar{u} \cdot \bar{w} \] This corresponds to a logic circuit where both \( u \) and \( w \) are passed through NOT gates and then ANDed together — as shown in option (A).