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3.6 AND AND OR OPERATORS AND THEIR MIXED-LOGIC CIRCUIT SYMBOLOGY 87

on the output of an inverting tri-state driver in the transfer mode indicates an inverted voltage
signal. Thus, if X is at LV, then X is at HV and vice versa.
Buffers, or line drivers as they are sometimes called, may be composed of a series of
inverters or gates used as inverters, or they may be simply a tri-state driver operated in the
transfer mode. Remember, it is the function of a line driver to boost and sharpen signals that
might otherwise degrade below switching levels or be distorted. The mixed logic circuit
symbols for buffers are given in Fig. 3.20a.

3.6 AND AND OR OPERATORS AND THEIR MIXED-LOGIC
CIRCUIT SYMBOLOGY

There are just two binary logic operations that underlie all of logic design and Boolean
algebra (after George Boole, 1815-1864, English mathematician). These are the AND and
OR operations. The following are the operator symbols (or connectives) that are used for
AND and OR:

(•) -> AND (+) -» OR

So, if one writes X • Y, XY, or (X)(F), it is read as X AND Y. Note that the AND operator (•) is
also called the Boolean product (or intersection) and may be represented by the alternative
symbol (A). Thus, X • Y = X A Y is the intersection or Boolean product of X and Y.
In contrast, X + Y is read as X OR Y. The operator (+) is often called the Boolean sum
(or union) and may be represented by the alternative symbol (v). Hence, X + Y = X v Y
is the union or Boolean sum of X and Y.
By using the two Boolean operators, an endless variety of Boolean expressions can be
represented. Simple examples are expressions such as
F = X + Y • Z and G = X • (Y + Z).

The first is read as F equals X OR (Y AND Z). In this expression the Boolean quantity
Y • Z must first be evaluated before it is "ORed" with X. The second expression is read as
G equals X AND (Y OR Z). In this case the quantity (Y + Z) must first be evaluated before
it can be "ANDed" with X. Thus, the hierarchy of Boolean operation is similar to that of
Cartesian algebra for multiplication and addition.

3.6.1 Logic Circuit Symbology for AND and OR
The meanings of the AND and OR operators (functions) are best understood in terms of their
logic circuit symbols. Shown in Fig. 3.9 are the distinctively shaped logic circuit symbols
commonly used to represent the AND and OR operators, which may have multiple inputs
and a single output. The functional descriptions of these symbols are stated as follows:

The output of a logic AND circuit symbol is active if, and only if, all inputs are active.

The output of a logic OR circuit symbol is active if one or more of the inputs are
active.