

Analogous behavior can be shown between Boolean algebra and mathematical algebra, and as a result, similar symbols and syntax can be used. For example, the following expressions hold true in math.  


This looks like the AND function allowing an analogy to be drawn between the mathematical multiply and the Boolean AND functions. Therefore, in Boolean algebra, A AND’ed with B is written A • B.  


Mathematical addition has a similar parallel in Boolean algebra, although it is not quite as flawless. The following four mathematical expressions hold true for addition.  


The first three operations match the OR function, and if the last operation is viewed as having a nonzero result instead of the decimal result of two, it too can be viewed as operating similar to the OR function. Therefore, the Boolean OR function is analogous to the mathematical function of addition.  


An analogy cannot be made between the Boolean NOT and any mathematical operation. Later in this chapter we will see how the NOT function, unlike AND and OR, requires its own special theorems for algebraic manipulation. The NOT is represented with a bar across the inverted element.  


The NOT operation may be used to invert the result of a larger expression. For example, the NAND function which places an inverter at the output of an AND gate is written as:  


Since the bar goes across A • B, the NOT is performed after the AND. Let’s begin with some simple examples. Can you determine the output of the Boolean expression 1 + 0 + 1? Since the plussign represents the OR circuit, the expression represents 1 or 0 or 1.  


Since an ORgate outputs a 1 if any of its inputs equal 1, then 1 + 0 + 1 = 1.  
The twoinput XOR operation is represented using the symbol ?, but it can also be represented using a Boolean expression. Basically, the twoinput XOR equals one if A = 0 and B = 1 or if A = 1 and B = 0. This gives us the following expression.  


The next section shows how the Boolean operators •, +, ?, and the NOT bar may be combined to represent complex combinational logic. 
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