5.2: Modeling Associative Operations in Logisim

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To demonstrate which Boolean operators are associative, the first step is to write equations for each operator which implement two associative ways to implement an expression, and see if the results are equivalent or not. A first equation for the AND operation could be the following: (((A * B) * C)*D). In this equation, the results of the output from each AND gate serially feeds the inputs to the AND next gate, where it is matched with the next input. This is shown in Figure \(\PageIndex{1}\) below:

Screen Shot 2020-06-26 at 6.06.55 PM.png — Figure \(\PageIndex{1}\): Serial AND circuit

A second equation, (A * B) * (C * D), does the first two AND operations in parallel, and then combines the results, as shown in Figure \(\PageIndex{2}\) below:

Screen Shot 2020-06-26 at 6.08.00 PM.png — Figure \(\PageIndex{2}\): Parallel AND circuit

These two circuits can be implemented in Logisim, and the results used to fill in the table for Exercise 2 in Section 5.5. This will show that for these two equations, the AND operator is associative. Changing the Boolean operator by inserting a different gate will give results for the rest of the table.