Question

Find the prime implicants of the following Boolean expressions by using the consensus method. (1) \(x y z+\bar{x} y+\bar{z} y+\bar{x} \bar{y}\) (3) \((\mathrm{x}+\mathrm{y})(\mathrm{y}+\mathrm{z})(\overline{\mathrm{z}}+\overline{\mathrm{x}})(\overline{\mathrm{x}}+\overline{\mathrm{y}}+\mathrm{z})(\mathrm{x}+\mathrm{y}+\mathrm{z})\)

Short answer

In summary, the prime implicants for the given Boolean expressions using the consensus method are: 1) \(x y z+\bar{x} y\): prime implicants are \(xyz\) and \(\bar{x}y\). 3) \((x+y)(y+z)(\bar{x}+z)\): prime implicants are \(x+y\), \(y+z\), and \(\bar{x}+z\).

Step-by-step solution

Identify the terms

For the first expression, we have four terms: \(xyz\), \(\bar{x}y\), \(\bar{z}y\), and \(\bar{x}\bar{y}\).

Apply consensus theorem

Apply consensus theorem on different pairs of terms to eliminate redundant terms. Consensus theorem for any three variables x, y, and z is given by: \(xyz + \bar{x}y + \bar{z}y = xyz + \bar{x}y\). Now, let's apply the consensus theorem to the terms we identified in step 1. \(x y z+\bar{x} y+\bar{z} y=\) \(x y z+\bar{x} y\)

Final expression

After applying the consensus theorem, the simplified expression becomes: \(x y z+\bar{x} y\) Thus, the prime implicants for expression (1) are \(xyz\) and \(\bar{x}y\). 3) \((x+y)(y+z)(\bar{z}+\bar{x})(\bar{x}+\bar{y}+z)(x+y+z)\)

Identify the sub-expressions

For the second expression, we have five sub-expressions: \((x+y)\), \((y+z)\), \((\bar{z}+\bar{x})\), \((\bar{x}+\bar{y}+z)\), and \((x+y+z)\).

Apply simplification rules

Apply simplification rules like distributive law, consensus theorem, and idempotent law to eliminate redundant expressions and simplify the given Boolean expression. The given expression can be rewritten using distributive law as: \((x+y)(y+z)(\bar{z}+\bar{x})(\bar{x}+\bar{y}+z)(x+y+z) =\) \((x+y)(y+z)(\bar{z}+\bar{x})(\bar{x}+z)(x+y+z)\) Now, applying consensus theorem to \((x+y)(y+z)(\bar{z}+\bar{x})(\bar{x}+z)(x+y+z)\), we get: \((x+y)(y+z)(\bar{z}+\bar{x})(\bar{x}+z)(x+y+z) = (x+y)(y+z)(\bar{x}+z)\)

Final expression

After applying simplification rules, the simplified expression becomes: \((x+y)(y+z)(\bar{x}+z)\) Thus, the prime implicants for expression (3) are \(x+y\), \(y+z\), and \(\bar{x}+z\).

Key concepts

Understanding Boolean Expressions

Boolean expressions are the backbone of digital logic and computer science. They are mathematical expressions that result in either a true or false outcome. In the context of digital electronics, these two outcomes are often represented as 1 for true and 0 for false. Boolean expressions are typically composed of variables and logical operators like AND (conjunction), OR (disjunction), NOT (negation), and sometimes XOR (exclusive or), among others.

For instance, an expression such as \(xyz + \bar{x}y + \bar{z}y + \bar{x}\bar{y}\) contains variables (x, y, z) and operations (AND, OR, NOT). Each term like \(xyz\) stands for a conjunction of variables - it is true only if x, y, and z are all true. A bar over a variable indicates NOT, which inverts the truth value of the variable. Simplifying Boolean expressions typically involves reducing them to a form with the fewest terms or variables, which is essential in designing efficient circuits.

Applying the Consensus Theorem

The consensus theorem is a simplification rule in Boolean algebra that can dramatically reduce the complexity of Boolean expressions without changing their outcomes. For any three Boolean variables x, y, and z, the consensus theorem states: \(xyz + \bar{x}y + \bar{z}y = xyz + \bar{x}y\). This theorem effectively tells us that under certain conditions, some terms in a Boolean expression are redundant and can be removed without affecting the overall result.

When you identify terms in an expression that fit the pattern of the consensus theorem, you can apply it to eliminate one of the terms, thereby simplifying the expression. In our exercise, applying the consensus theorem to the first expression reduced it, thereby revealing the prime implicants: the essential components of the simplified expression. These prime implicants are crucial in logic circuit design as they represent the simplest form of the original logic.

Leveraging Simplification Rules

Simplification rules in Boolean algebra, such as the distributive law, consensus theorem, and idempotent law, are used to reduce the complexity of Boolean expressions. These rules help us to transform expressions into their most compact form, removing redundancies and unnecessary parts. The motivation behind this simplification is not purely academic; it has practical applications in optimizing digital circuits and making computations more efficient.

For example, in the second part of the exercise, we simplified a more complex Boolean expression by first rewriting it using the distributive law, then applying the consensus theorem, and possibly other rules like the idempotent law, which states that \(x + x = x\) and \(xx = x\). These simplification rules facilitate the identification of prime implicants, which inform us about the minimum necessary conditions to replicate the original expression's function in a logical circuit or a computational algorithm.