Question

Draw a combinatorial circuit for each of the following boolean expressions: (a) \((x \wedge y) \vee \neg z\) (b) \((x \wedge y) \vee(\neg x \wedge y)\) (c) \(\neg(\neg x \vee y) \vee(x \wedge z)\) (d) \(((x \wedge y) \vee(y \wedge z)) \vee \neg z\) (e) \((x \vee \neg(x \vee y)) \vee(\neg x \wedge \neg y)\)

Short answer

(a) Combine AND and NOT with OR. (b) OR outputs from two AND gates. (c) OR result of NOT and AND. (d) OR from two ANDs with NOT z. (e) OR NOT result with AND of negated inputs.

Step-by-step solution

Identify the logic gates

Begin by identifying the logic gates required for the Boolean expressions. Use AND (AND), OR (OR), and NOT (NOT) gates to represent the operations: \((\wedge)\) is represented by an AND gate, \((\vee)\) by an OR gate, and \((eg)\) by a NOT gate.

Expression (a) Analysis and Mapping

For expression (a) \((x \wedge y) \vee eg z\):1. Use an AND gate for \(x \wedge y\).2. Use a NOT gate for \(z\) to get \(eg z\).3. Combine the outputs of the AND gate and the NOT gate with an OR gate.

Expression (b) Analysis and Mapping

For expression (b) \((x \wedge y) \vee(eg x \wedge y)\):1. Use two AND gates: one for \(x \wedge y\) and another for \(eg x \wedge y\). 2. Use a NOT gate to obtain \(eg x\).3. Connect the output of the NOT gate to one input of the second AND gate.4. Combine the outputs of both AND gates with an OR gate.

Expression (c) Analysis and Mapping

For expression (c) \(eg(eg x \vee y) \vee(x \wedge z)\):1. Use a NOT gate followed by an OR gate to evaluate \(eg x \vee y\).2. Apply another NOT gate to the output to get \(eg(eg x \vee y)\).3. Use an AND gate for \(x \wedge z\).4. Use an OR gate to combine the results of the second NOT and the AND gate.

Expression (d) Analysis and Mapping

For expression (d) \(((x \wedge y) \vee(y \wedge z)) \vee eg z\):1. Use two AND gates: one for \(x \wedge y\) and another for \(y \wedge z\).2. Use an OR gate to combine the outputs of these AND gates.3. Use a NOT gate to get \(eg z\).4. Finally, combine the outputs of the OR gate from step 2 and the NOT gate using another OR gate.

Expression (e) Analysis and Mapping

For expression (e) \((x \vee eg(x \vee y)) \vee(eg x \wedge eg y)\):1. Use an OR gate for \(x \vee y\).2. Use a NOT gate for the output to get \(eg(x \vee y)\).3. Use a second OR gate for \(x \vee eg(x \vee y)\).4. Use two NOT gates to get \(eg x\) and \(eg y\).5. Use an AND gate for \(eg x \wedge eg y\).6. Finally, combine the outputs from step 3 and step 5 with an OR gate.

Key concepts

Boolean Expressions

Boolean expressions are the foundation of combinational circuit design. They represent logical statements that can either be true or false.

- Complex logical statements can be broken into simpler components using standard operators like AND, OR, and NOT. - These operators provide a method to manipulate and evaluate expressions.

Boolean expressions are often used to perform calculations and represent information in digital circuits. Understanding how to convert these expressions into a workable circuit layout is essential for creating efficient digital systems. It assists in simplifying and optimizing the way circuits are constructed.

Logic Gates

Logic gates are the building blocks of digital circuits. They perform basic logical functions that determine the flow of electrical signals.

- Each type of logic gate corresponds to a different Boolean operation. - For example, an AND gate only allows the signal to pass through if all conditions are true. An OR gate permits the signal if any condition is true. - A NOT gate, on the other hand, inverts the signal, changing true to false or false to true.

Designers use these gates to compile larger, more complex circuits from simple logical operations. Being familiar with each type of gate and their functions is crucial for anyone looking to grasp digital circuit design.

AND, OR, NOT Gates

The AND, OR, and NOT gates are the primary elements used to interpret and execute Boolean expressions.

• AND Gate: Represents the logical conjunction in Boolean algebra. It outputs true only if all its inputs are true.
• OR Gate: Represents the logical disjunction. It outputs true if at least one of its inputs is true.
• NOT Gate: Represents logical negation. It inverts its input, changing a true to false, and vice versa.

These gates can be combined in various configurations to solve more complex Boolean expressions and are fundamental in building both simple and composite circuits.

Circuit Mapping Steps

Circuit mapping is the process of translating Boolean expressions into a schematic representation using logic gates. This involves several methodical steps:

- **Identify the Gates Required**: Determine which gates (AND, OR, NOT) are needed to represent each part of your expression. - **Break Down the Expression**: Dissect a complex expression into smaller, manageable segments. Each segment corresponds to a particular gate or set of gates.

Follow the defined order of operations within the expression, such as handling parentheses and NOT operations first, akin to arithmetic hierarchy. By synthesizing these steps into a physical or diagrammatic circuit, you create a blueprint that gives life to the logical function represented by the Boolean expression. This process is key in transforming theoretical calculations into practical, real-world applications.