Question

Construct a switching circuit for the following Boolean expression: (a) \((\underline{A B C}+C \underline{B}) D+A\) (b) \((\mathrm{A}+\mathrm{B}+\mathrm{C})(\underline{\mathrm{B}}+\mathrm{C})\)

Short answer

For Boolean expression (a) \((\underline{A B C}+C \underline{B}) D+A\), construct a switching circuit as follows: 1. Connect \(A\), \(B\), and \(C\) to a 3-input AND gate and its output to a NOT gate. 2. Connect \(C\) to an AND gate and \(B\) to a NOT gate, which then connects to the same AND gate. 3. Connect the output from steps 1 and 2 to a 2-input OR gate. 4. Connect this output to an AND gate along with input \(D\). 5. Connect the output from step 4 and input \(A\) to a 2-input OR gate. The output of this gate is the required result. For Boolean expression (b) \((\mathrm{A}+\mathrm{B}+\mathrm{C})(\underline{\mathrm{B}}+\mathrm{C})\), construct a switching circuit as follows: 1. Connect inputs \(A\), \(B\), and \(C\) to two 2-input OR gates in series so that their outputs are OR'ed together. 2. Connect input \(B\) to a NOT gate. 3. Connect the output of step 2 to a 2-input OR gate along with input \(C\). 4. Connect the output from steps 1 and 3 to a 2-input AND gate. The output of this gate is the required result.

Step-by-step solution

Identify the Boolean expression (a)

The first Boolean expression we need to create a switching circuit for is: \((\underline{A B C}+C \underline{B}) D+A\).

Gates and connections for (a)

To create the switching circuit for this expression, we will break down its parts and identify the required logic gates: 1. \(A\), \(B\), and \(C\) is a triple AND operation and we have to negate it: \(\underline{A B C}\). We will use an AND gate and a NOT gate for this. 2. \(C\) and negated \(B\): \(C \underline{B}\). We will need an AND gate and a NOT gate for this. 3. Results of 1 and 2 are OR'ed together: \((\underline{A B C}+C \underline{B})\). We will use an OR gate. 4. The result in step 3 is AND'ed with \(D\): \((\underline{A B C}+C \underline{B})D\). This will require an AND gate. 5. Finally, the outcome in step 4 is OR'ed with \(A\): \((\underline{A B C}+C \underline{B}) D+A\). We will use an OR gate.

Constructing the switching circuit for (a)

Now, following the steps we identified in the Gates and connections for (a): 1. Connect \(A\), \(B\), and \(C\) to a 3-input AND gate and its output to a NOT gate. 2. Connect \(C\) to an AND gate and \(B\) to a NOT gate, which then also connects to the same AND gate (used to make \(C \underline{B}\)). 3. Connect the output from step 1 and step 2 to a 2-input OR gate. 4. Connect the output from step 3 to an AND gate along with input \(D\). 5. Connect the output from step 4 and input \(A\) to a 2-input OR gate. The outcome of this gate is the required output of the Boolean expression (a).

Identify the Boolean expression (b)

The second Boolean expression we need to create a switching circuit for is: \((\mathrm{A}+\mathrm{B}+\mathrm{C})(\underline{\mathrm{B}}+\mathrm{C})\).

Gates and connections for (b)

To create the switching circuit for this expression, we will break down its parts and identify the required logic gates: 1. \(A\), \(B\), and \(C\) are OR'ed together: \((A+B+C)\). We will use two OR gates for this. 2. Negate \(B\): \(\underline{B}\). We will need a NOT gate. 3. The result of step 2 is OR'ed with \(C\): \((\underline{B}+C)\). We will use an OR gate. 4. Finally, the outcomes of step 1 and step 3 are AND'ed together: \((A+B+C)(\underline{B}+C)\). This will require an AND gate.

Constructing the switching circuit for (b)

Now, following the steps we identified in the Gates and connections for (b): 1. Connect inputs \(A\), \(B\), and \(C\) to two 2-input OR gates in series so that their outputs are OR'ed together. 2. Connect input \(B\) to a NOT gate. 3. Connect the output of step 2 to a 2-input OR gate along with input \(C\). 4. Connect the output from step 1 and step 3 to a 2-input AND gate. The outcome of this gate is the required output of the Boolean expression (b).

Key concepts

Boolean Algebra

Boolean algebra is a mathematical framework that deals with binary variables and logical operations. It uses only two values: true (1) and false (0), which correspond perfectly with the on and off states in an electronic circuit. The fundamental operations of Boolean algebra are conjunction (AND), disjunction (OR), and negation (NOT).

Understanding Boolean algebra is crucial for designing and interpreting digital circuits. It allows us to transform and simplify complex logic expressions, such as the example exercise, where expressions like can be systematically reduced using Boolean laws and theorems like De Morgan's Theorems, Distributive Law, and Associative Law. These reductions are the preliminary steps required before proceeding to the circuit construction phase.

Logic Gates

Logic gates are the building blocks of digital circuits. They are devices that perform basic logical functions defined by Boolean algebra. There are seven primary gates: AND, OR, NOT, NAND, NOR, XOR and XNOR. Each gate has a distinct symbol and performs a specific operation on one or more input signals to produce an output.

In our exercise example, various logic gates including AND, OR, and NOT were used to create switching circuits. When representing Boolean expressions as diagrams, these gates serve as visual representations of the logical operations taking place. For instance, the expression corresponds to a sequence of gates that express the logical operations described by the expression, ultimately allowing for the construction of circuits that perform diverse functions based on input combinations.

Circuit Construction

Circuit construction is the physical realization of a logic gate arrangement that represents a specific Boolean expression. The process begins with a logical diagram, which visually maps out the connections between gates as defined by the Boolean expression. Then, physical components such as integrated circuits or transistors are connected according to this diagram to create the actual electronic circuit.

Following the step-by-step methodology provided in our exercise solution is key for successful circuit construction. For instance, the solution first outlines the required gates and connections and then guides through connecting inputs to the correct gates in a sequential order. Each connection corresponds to an operation in the Boolean expression, and the accuracy of these connections directly impacts the functionality of the completed circuit.

Boolean Expressions

Boolean expressions are a series of symbols in Boolean algebra that collectively describe logic operations to be performed on binary inputs. These expressions follow specific syntax rules and use variables along with logical operators to form complex logical functions.

Understanding how to interpret and manipulate Boolean expressions is fundamental in designing digital circuits. In the example exercise, expressions such as are blueprints for the logic sequences required in the circuit. The ability to read and simplify these expressions using Boolean theorems enhances the efficiency of the circuit and guides the construction process from a theoretical concept to a functional digital device.