Page 59 - Bebop to The Boolean Boogie An Unconventional Guide to Electronics Fundamentals, Components, and Processes

40 ChaptevFiwe

The AND and OR representations shown here are the abstract equivalents

of our original switch examples. In the case of the AND, the output is only
TRUE if both a b are TRUE; in the case of the OR, the output is TRUE if either
a b are TRUE. In fact, the OR should more properly be called an inclusive-OR,
because the TRUE output cases include the case when both inputs are TRUE.

Contrast this with the exclusive-OR, or XOR, where the TRUE output cases
exclude the case when both inputs are TRUE.

NAND, NOR, and XNOR Functions

Now consider the effect of appending a NOT function to the output of the
AND function (Figure 5-6).

This combination of functions occurs frequently in designs. Similarly, the
outputs of the OR and XOR functions are often inverted with NOT functions.
This leads to three more primitive functions called NAND (NOT-AND),

NOR (NOT-OR) and NXOR (NOT-XOR).6 In practice, however, the NXOR
is almost always referred to as an XNOR (exclusive-NOR) (Figure 5-7).7

The bobbles on their outputs indicate that these are inverting functions.
One way to visualize this is that the symbol for the NOT function has been
forced back into the preceding symbol until only the bobble remains visible.
Of course, if we appended a NOT function to the output of a NAND,
we’d end up back with our original AND function again. Similarly, appending
a NOT to a NOR or an XNOR results in an OR and XOR, respectively.

Not a Lot

And that’s about it. In reality there are only eight simple functions (BUF,
NOT, AND, NAND, OR, NOR, XOR, and XNOR) from which everything
else is constructed. In fact, some might argue that there are only seven core
functions because you can construct a BUF out of two NOTs, as was discussed
earlier.

6 In conversation, the terms NAND and NOR are pronounced as single words to rhyme with
“band” and “door”, respectively.
7 In conversation, the term XNOR is pronounced “X-NOR,” that is, spelling the letter “X”
followed by “NOR to rhyme with “door.”