Problem 5 - Discrete Mathematics with Applications

Question

Simplify each boolean expression using the laws of boolean algebra. $$x^{\prime} y z+x^{\prime} y^{\prime} z^{\prime}+x^{\prime} y z^{\prime}+x^{\prime} y^{\prime} z$$

Step-by-Step Solution

Answer

The simplified Boolean expression is \(x^{\prime} y z + x^{\prime} y^{\prime} z^{\prime} + x^{\prime} y z^{\prime}\).

Step 1: Apply Consensus Theorem

Use the Consensus Theorem to simplify the expression, which states that \(x y + x^{\prime} z + y z = x y + x^{\prime} z\). \[ x^{\prime} y z + x^{\prime} y^{\prime} z^{\prime} + x^{\prime} y z^{\prime} + x^{\prime} y^{\prime} z = x^{\prime} y z + x^{\prime} y^{\prime} z^{\prime} + x^{\prime} y z^{\prime} (1 + y^{\prime}) \]

Step 2: Simplify the Resulting Expression

Simplify the expression further by applying the Dominance Law and the Idempotency Law, which state that \(1+x = 1\) and \(x+x=x\) respectively. \[ x^{\prime} y z + x^{\prime} y^{\prime} z^{\prime} + x^{\prime} y z^{\prime} (1 + y^{\prime}) = x^{\prime} y z + x^{\prime} y^{\prime} z^{\prime} + x^{\prime} y z^{\prime} (1) \]

Step 3: Final Simplification

Rewrite the expression by eliminating the term \(x^{\prime} y z^{\prime} (1)\), since multiplying by \(1\) does not change the value of the term. \[ x^{\prime} y z + x^{\prime} y^{\prime} z^{\prime} + x^{\prime} y z^{\prime} = x^{\prime} y z + x^{\prime} y^{\prime} z^{\prime} + x^{\prime} y z^{\prime} \] The simplified Boolean expression is \(x^{\prime} y z + x^{\prime} y^{\prime} z^{\prime} + x^{\prime} y z^{\prime}\).

Key Concepts

Consensus Theorem

The Consensus Theorem in Boolean algebra is a simplification technique that helps reduce the complexity of expressions. It can be a lifesaver when dealing with elaborate logic circuits. In essence, the theorem states that for any Boolean variables x, y, and z, the expression \(x y + x' z + y z = x y + x' z\) holds true.

This implies that you can remove the \(y z\) term without altering the outcome of the expression. Visually, imagine that there's an agreement or 'consensus' between the terms \(x y\) and \(x' z\), making the \(y z\) term redundant.

To apply this theorem, look for terms in the expression that have one variable in common and the other two variables as complementary pairs. Once identified, you can safely omit the redundant term to simplify the expression. It's a powerful tool that often makes circuit designs more efficient and cost-effective.

Dominance Law

Dominance Law simplifies a Boolean expression by allowing us to replace certain parts of the expression with a logical constant that 'dominates' the outcome. For instance, in any logical OR operation, if one of the inputs is 1, the result is always 1, regardless of the other input. Mathematically, this is expressed as \(x + 1 = 1\).

Similarly, in a logical AND operation, if one of the inputs is 0, the result is always 0, which can be captured by the equation \(x \cdot 0 = 0\). The similarity of Dominance Law to certain truths we know in arithmetic (e.g., any number multiplied by zero is zero) makes it intuitively easy to grasp.

This law becomes incredibly handy when simplifying complex Boolean expressions, as it allows significant reduction just by spotting a 'dominating' element. In logical design, it helps simplify the number of gates required and, consequently, the overall costs and power consumption of electronic devices.

Idempotency Law

The Idempotency Law is another simplification principle in Boolean algebra that states that an element ORed or ANDed with itself yields the same element. Symbolically, for any Boolean variable x, it can be expressed as \(x + x = x\) and \(x \cdot x = x\). This means, in a logical OR operation, if both inputs are the same, the output will be that same input. Similarly, for an AND operation, the same rule applies.

While this may seem trivial, it's quite powerful when it comes to simplifying Boolean expressions. By recognizing and eliminating redundant terms, we can streamline complex expressions into something far more manageable. Essentially, it promotes a sense of 'less is more', where duplications are dropped to achieve elegance and simplicity in both symbolic and physical logic representations.

Idempotency is akin to stating that adding zero to a number or multiplying a number by one does not change its value. It reflects the same type of simplification but in the realm of Boolean logic.