How to Solve Logic Diagram to Boolean Expression Problems?

How to Solve Logic Diagram to Boolean Expression Problems?

In this type of problem, a gate-based circuit diagram or schematic is provided, and the task is to determine the equivalent Boolean expression at the output. To solve the logic diagram and derive its corresponding mathematical expression, we will follow a systematic step-by-step approach.

Evaluating a Logic Circuit Example:

Now that we’ve outlined the steps for analyzing a logic circuit, let’s apply them to a specific example. We’ll walk through a logic diagram, starting by identifying its inputs, outputs, and the direction of signal flow. Then, we’ll evaluate each gate one by one to trace how the signals propagate through the circuit and arrive at the final output. This step-by-step breakdown will help clarify how the logic components work together to produce the desired result.

Step 1: Spot the input and output signals, and note the direction of signal flow.

Let’s begin by identifying the input and output signals of the circuit. In this example, the circuit includes three input signals—A, B, and C—and produces a single output signal, labeled Z. By convention, the diagram flows from left to right, which is typical in digital circuit design. However, keep in mind that some diagrams may flow from right to left, depending on how they’re presented.

Immediately after the input stage, you’ll notice that each input line branches off to a NOT gate, creating a parallel path. The output from each NOT gate is the complement of the respective input signal—\(\mathsf{A}\) becomes \(\mathsf{\overline{A}}\) or \(\mathsf{A’}\), \(\mathsf{B}\) becomes \(\mathsf{\overline{B}}\) or \(\mathsf{B’}\), and so on.

To aid readability, the inputs and their complements are color-coded, with \(\mathsf{A}\) and \(\mathsf{B}\) connected to the AND gate, and the NOT of signal \(\mathsf{B, \overline{B}}\) and \(\mathsf{C}\) connected to the OR gate.

This design technique is often used early in digital logic courses to help students recognize the relationship between an input and its complement. Later in the course, these input-complementing NOT gates are usually omitted visually—instead, the complemented form of the signal is directly represented at the gate input where needed. This saves space and removes unnecessary clutter. Circuit diagram without dedicated lines for the signal and its complement is shown here:

Step 2: Trace each gate from input to output, building the Boolean expression step by step.

Now that we’ve identified the inputs and output(s), we may also encounter intermediate signals—connections that link the output of one gate to the input of another. These signals aren’t primary inputs or final outputs, and they may or may not be labeled depending on the circuit’s complexity or the problem’s focus.

To solve such problems, begin by identifying each gate in the circuit, starting from the input side. Write down the output of each gate, then treat that output as the input for the next gate, following the circuit’s connectivity. Repeat this process until you reach the final output.

In this example, the circuit contains three gates: one AND gate and two OR gates. For clarity, we can label them as a1 (AND gate), and o1 and o2 (OR gates). While these labels aren’t required in the final solution, they help simplify the analysis—especially when solving a circuit for the first time.

Here’s how the logic flows:

• Inputs \(\mathsf{A}\) and \(\mathsf{B}\) connect to the AND gate a1, producing the output \(\mathsf{AB}\).
• Inputs \(\mathsf{\overline{B}}\) and \(\mathsf{C}\) connect to the OR gate o1, producing the output \(\mathsf{\overline{B}+C}\).
• These two intermediate outputs—\(\mathsf{AB}\) and \(\mathsf{\overline{B}+C}\)—then feed into the second OR gate o2, whose output is \(\mathsf{\left(AB\right)}\) + \(\mathsf{\left(\overline{B}+C\right)}\).

Since this is the final gate in the circuit, the output signal \(\mathsf{Z}\) is equal to the output of o2. Therefore, the Boolean expression for the circuit is: \(\mathsf{\left(AB\right)+\left(\overline{B}+C\right)}\). It is best to keep the individual outputs enclosed in brackets for clarity.

This section focused on Logic Diagram to Boolean Expression, guiding you through its core concepts and practical examples. Whether you’re converting diagrams into Boolean expressions, tracing output values, or interpreting mathematical logic, each walk through is designed to build clarity and confidence. If you’re curious about other types of logic problems—like constructing circuits from mathematical notation, evaluating outputs using symbolic expressions, or calculating output latency across gate paths—you’ll find dedicated sections that break down each approach step by step. Dive into these categories to strengthen your understanding and navigate seamlessly between different problem-solving strategies for simple gate-based circuits.