Discrete Math with SageMath: Learn math with open-source software

Logic gates can be combined to create more complex circuits that perform specific tasks. By linking gates together, we can create circuits that process multiple inputs to produce a desired output. For example, combining an AND gate and a NOT gate results in a NAND gate, which inverts the output of the AND gate. More complex circuits, such as half-adders and multiplexers, are built by combining basic gates in strategic ways.

Let’s look at a circuit:

We will evaluate this circuit by setting True for \(X\text{,}\) \(Y\text{,}\) and False for \(Z\) below using Sage.

Boolean algebra provides a way to simplify complex logic circuits. By using Boolean algebra rules, you can take a complicated circuit and reduce it to a simpler form without changing its functionality.

Here’s a practical example. Consider the following Boolean expression, which combines several gates:

Truth tables are a visual way to represent how inputs to a logic circuit map to its outputs. For each possible combination of inputs, the table shows the corresponding outputs, making it easier to analyze and understand the behavior of the circuit.

Let’s create a truth table for the simplified circuit.

Here, we will show the intermediary steps to find the final output of the function.