Boolean Algebra - Mathematics and Statistics for the Environment - page 214

• The power supply fails.
• The switch is open.
• The fuse is blown.
• Light 1 and light 2 are burned out.
Each of the four sets above represents a cut set, because, if all of the events in any of the sets occur,
then the end event (room being dark) will occur. For the example in Figure 9.1, the end event will
not occur if no events in either of the following sets occur:
• The power supply fails, the switch is open, the fuse is blown, and light 1 is burned out, or
• The power supply fails, the switch is open, the fuse is blown, and light 2 is burned out.
Each of the two sets above represents a path set, because, if none of the events in either of the sets
occurs, then the end event (room being dark) cannot occur.
9.1.2 K ey t erms
Boolean variable is usually represented by a capital letter, representing a distinct event or fact.
A, B, and C are Boolean variables.
(+) is the logic OR operator.
(•)isthelogicANDoperator.
9.2 TECHNICAL OVERVIEW
Boolean algebra variables are usually depicted by capital letters representing distinct events or
facts. For example, we may let A represent the event that the belt on a certain machine pulley system
breaks. If this occurs, we say that A = T or A is true. If the event fails to occur, we say A = F or A is
false. Of course, there must be some finite time during which the system is under consideration, and
there is a probability associated with event A , although it is often unknown.
The most obvious way to simplify Boolean expressions is to manipulate them in the same way
as normal algebraic expressions are manipulated. The manipulations, expressed as true or false (i.e.,
occurrence or nonoccurrence), are two modes of a Boolean algebra variable. These modes can be
formed into functions, as a combination of Boolean algebra variables. With regard to logic relations
in digital forms, a set of rules for symbolic manipulation is needed in order to solve for the unknowns.
A set of rules formulated by the originator of Boolean expressions, George Boole, described
certain propositions whose outcomes correspond to the true or false relationship described above.
Again, with regard to digital logic, these rules are used to describe circuits or possibilities whose
state can be either 1 (true) or 0 (false). In order to fully understand this, the AND, OR, and NOT
operators should be appreciated. A number of rules can be derived from these relations, as Table
9.1 demonstrates.
Table 9.2 shows the basic Boolean laws we are concerned with in this text. Note that every law
has two expressions, (a) or (b). This is known as duality . These are obtained by changing every
AND(•)toOR(+),everyOR(+)toAND(•),andall1'sto0'sand vice versa . In most cases, it has
becomeconventionaltodroptheANDsymbol(•);thatis, A • B is simply written as AB .
9.2.1 C ommutative l aW
The commutative law for Boolean algebra is not much different than the commutative law for
numeric algebra. For both the AND or OR logic operations, the order in which a variable is pre-
sented will not affect the outcome. So the first equation, A + B = B + A , may be read as “ A OR B
equals B OR A .” It follows that AB + BA also reads as “ A AND B equals B AND A .”