5.1: Introduction

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This chapter looks at which Boolean operators are associative. Associative operations allow arbitrary groupings of the operations. For example, addition is associative. We can show this with the following two equations, which are equal:

x = (2 + (3 + (4+5))) = 14

x = (2 + 3) + (4 + 5) = 14

Subtraction is not associate, as can be seen in the equations below:

x = (2 - (3 - (4 - 5))) = -2

x = ((2 - 3) - (4 - 5)) = 0

These examples show that subtraction is not associative, and while they do not prove that addition is associative, they are illustrative that addition is associative at least for this example. The proof that addition is associative is not really of interest in this text, and can easily be found in an online search.

Like arithmetic operators, Boolean operators can also be associative, commutative, and distributive. This chapter will create circuits which will demonstrate the associative property for Boolean operators. The exercises at the end of the chapter allow the reader to further explore these properties.