Boolean algebra

The logical structure of a fault tree can be described in terms of boolean algebraic equations. Some specific prerequisites to the application of this methodology are as follows. 

Table 2.1-1 compares the ordinary algebra of continuous variables with the Boolean algebra of 1 s and Os. This table uses the symbols and -h for the operations of intersection (AND) and union (OR) which mathematicians represent by n and u respectively. The symbols and -f which are the symbols of multiplication and addition, are used because of the similarity of their use to AND and OR in logic. 

In the table, the rules of commutation, association and distribution are the same for both algebras. Idempotency, unique to Boolean algebra, relates to redundancy. Having "A and A is the same as only having A A or A is superfluous and equals A. Complementation is introduced in the next rule. The universe is represented by 1 . Completeness includes every thing in this world and not in this world, hence A +A = 1 where not A is A 1 -A which is the meaning of complementation. With this understanding, it is impossible to be both A and not A. Similarly A or not A is complete (the universe). Under unity, A is included in the universe (1) so A-f 1 = 1 For this... 

Systems may be modeled using Boolean algebra with two-.state representation of component operability. Systems are not usually modeled as equations is not usually done because of preference by engineers for the more schematic methods that are presented in the next chapter. 

The term "min" is readily understood when Boolean algebraic reduction is applied to the logic equation, because terms like A + A C collapse to A, thereby affording considerable simplification. Tlie reason for this is if A causes failure, then additional failures are superfluous. 

Accident Sequence Quantification estimates the IE frequency. Specifically, the plant model built in the Step 2 is quantified by data from Step 3 according to Boolean algebra. Quantification may be a point-value calculation in which all parameters are delermimsiic, or as uncertain values known by their distribution function. 

The algebra of sets-BooIean algebra-governs tlie way in wliich sets can be manipulated to form equivalent sets. The principal Boolean algebra laws used for tliis purpose are as follows. 

If the letter symbols for sets are replaced by numbers, tlie commutative and associative laws become familiar laws of aritlimetic. In Boolean algebra tlie first of tlie two distributive laws, Eq. (19.3.5), lias an analogous counterpart in arithmetic. Tlie second, Eq. (19.3.6), does not. In risk analysis. Boolean algebra is used to simplify e. pressions for complicated events. For example, consider tlie event... 

Computer science and algebra The symbolic system of mathematical logic called Boolean algebra represents relationships between entities either ideas or objects. George Boole of England formulated the basic rules of the system in 1847. The Boolean algebra eventually became a cornerstone of computer science. 

Exercise Verify the equivalence of the two definitions of a Boolean algebra. 

Exercise Draw diagrams to illustrate the notions of fcontainment, union, and intersection, and the properties of a Boolean algebra. Hence, a and b are independent if, and only if... 

In order to bridge the gap between the discretized micro- and macro-worlds, averaging of the variables is necessary. Macroscopic variables in the N-S equation, are the density p and the momentum I, which are functions of the lattice space vector r and time t. The local density p is the summation of the average number of particles travelling along each of six (hexagonal) directions, with velocity c. Multiplication of the density p by the velocity vector u equals linear momentum (I = pu). Boolean algebra is applied for the expressions of the discretized variables density and momentum, respectively, as follows ... 

Since ferroelectricity was discovered in 1921 it has been obvious to many scientists and engineers that the two stable polarization states +P and P could be used to encode the 1 and 0 of the Boolean algebra that forms the basis of memory and logic circuitry in all modem computers. Yet until very recently this has been unsuccessful. In fact, although ferroelectric materials are used in a wide variety of commercial devices, it has until now always been the case that some other property of the material - especially pyroelectricity or piezoelectricity - is the characteristic actually employed. Ironically, no devices using ferroelectrics have actually required ferroelectricity to work. 

Applying tlie Boolean algebra associative law and distributive law, given in Eqs. 

Gregg JR. Ones and Zeros Understanding Boolean Algebra, Digital Circuits and the Logic of Sets. 1998. John Wiley Sons, Inc., New York. 

Digital devices such as computers work with sharply defined logical elements—bits which take the values of 1 (true) and 0 (false) i.e., their work is based on classical prepositional logic or its mathematical equivalent, Boolean algebra. The fuzzy nature of human reasoning, on the other hand, makes things much more flexible, but may lead to irreproducible results and hence be a source of error. For instance, two chemists may draw different conclusions about a structure derived from the same spectral information from the same unknown compound measured on the same instrument. In contrast, the classical logic of computers always leads to the... 

Boobies and gannets Boolean algebra Boric acid Botany Botulism... 

In this section we will summarize some of the most useful mathematical operations available in Excel. This section is merely for your information, just to give you an idea of what is available it is certainly not meant to be memorized. There are many more functions, not listed here, that are mainly used in connection with statistics, with logic (Boolean algebra), with business and database applications, with the manipulation of text strings, and with conversions between binary, octal, decimal and hexadecimal notation. 

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