# Discrete

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## References

• (Wordnet, 2009) ⇒ http://wordnet.princeton.edu/perl/webwn?s=discrete
• S: (adj) discrete, distinct (constituting a separate entity or part) "a government with three discrete divisions"; "on two distinct occasions"
• (Wikipedia, 2009) ⇒ http://en.wikipedia.org/wiki/Discrete_(signal)
• A discrete signal or discrete-time signal is a time series consisting of a sequence of quantities. In other words, it is a time series that is a function over a domain of discrete integers. Each value in the sequence is called a sample.
• Unlike a continuous-time signal, a discrete-time signal is not a function of a continuous argument; however, it may have been obtained by sampling from a continuous-time signal. When a discrete-time signal is a sequence corresponding to uniformly spaced times, it has an associated sampling rate; the sampling rate is not apparent in the data sequence, so may be associated as a separate data item.
• http://en.wiktionary.org/wiki/discrete
• Adjective
• 1. Separate; distinct; individual.
• 2. Something that can be perceived individually and not as connected to, or part of something else.
• 3. (electrical engineering) Having separate electronic components, such as individual resistors and inductors — the opposite of integrated circuitry.
• 4. (audio engineering) Having separate and independent channels of audio, as opposed to multiplexed stereo or quadraphonic, or other multi-channel sound.
• 5. (topology) Having each singleton subset open: said of a topological space or a topology.
• (Wikipedia, 2009) ⇒ http://en.wikipedia.org/wiki/Discrete_mathematics
• Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. Real numbers and rational numbers have the property that between any two numbers a third can be found, and consequently these numbers vary "smoothly". The objects generally studied in discrete mathematics – such as integers, graphs, and statements in logic[1] – do not vary smoothly in this way, but have distinct, separated values. [2] Discrete mathematics therefore excludes topics in "continuous mathematics" such as calculus and analysis. Alternatively, discrete mathematics can be characterised as the branch of mathematics dealing with countable sets [3] (including rational numbers but not real numbers), but there is no exact, universally agreed, definition of the term. [4] It is more what is excluded (the notions of a continuously varying quantity and related notions) than what is included that describes discrete mathematics. The term finite mathematics is sometimes applied to parts of the field of discrete mathematics, particularly areas relevant to business.
• Discrete mathematics has become popular in recent decades because of its applications to computer science. Concepts and notations from discrete mathematics are useful in studying and describing objects and problems in computer algorithms and programming languages, and have applications in cryptography, automated theorem proving, and software development.

Computer implementation is an important aspect of discrete mathematics. Here a computer is being used to verify statements in logic, which is helpful in developing software for safety-critical systems.

• The distinction between discrete mathematics and other mathematics is somewhat artificial as analytic methods are often used to study discrete problems and vice versa. Number theory in particular sits on the boundary between discrete and continuous mathematics