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Boolean algebra was introduced in \(1854\) by George Boole and has been very important for the development of computer science. It operates with variables which have the truth values true and false (sometimes denoted 1 and 0 respectively). In this chapter the main operations of Boolean algebra (conjunction (AND, \(\wedge \)), disjunction (OR, \(\vee \)) and negation (NOT, \(\lnot \))) are defined, and statements involving logical variables are studied with the aid of truth tables and Venn diagrams. Useful formulae involving logical variables are then discussed (De Morgan’s laws), along with the existential and universal quantifiers and their negation.

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An encryption method is presented with the novel property that publicly revealing an encryption key does not thereby reveal the corresponding decryption key. This has two important consequences: Couriers or other secure means are not needed to transmit keys, since a message can be enciphered using an encryption key publicly revealed by the intended recipient. Only he can decipher the message, since only he knows the corresponding decryption key. A message can be “signed” using a privately held decryption key. Anyone can verify this signature using the corresponding publicly revealed encryption key. Signatures cannot be forged, and a signer cannot later deny the validity of his signature. This has obvious applications in “electronic mail” and “electronic funds transfer” systems. A message is encrypted by representing it as a number M, raising M to a publicly specified power e, and then taking the remainder when the result is divided by the publicly specified product, n , of two large secret prime numbers p and q. Decryption is similar; only a different, secret, power d is used, where e * d = 1(mod (p - 1) * (q - 1)). The security of the system rests in part on the difficulty of factoring the published divisor, n .
Covering material suitable for a first year course in mathematics for computing science specialists, this text introduces mathematics in the context of its use in programming. The book also introduces the role of maths in programming, suitable for interested professionals in the software industry. Topics include: the basic mathematical language of sets; functions and relations; propositions and predicates. These are used in the context of various data types - natural numbers; characters and strings; generic types including stacks and trees. Functions are used to describe the effect of calculations, logic to describe what is required in a rigorous way, and other mathematical structures to provide the initial formal model of a "real" problem. Functions are used both to capture the effect of some calculation, and what is required from a calculation.
Contenido: Introducción a los conjuntos; Proposiciones lógicas; Predicados lógicos: Relaciones; Relaciones homogéneas; Funciones; Modelos matemáticos.
  • S Lang
Lang, S.: Algebra. Graduate Texts in Mathematics 211 (Revised third ed.), Springer-Verlag (2002)