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The system + =, + = has exactly one solution: x = 1, y = 2 The nonlinear system + =, + = has the two solutions (x, y) = (1, 0) and (x, y) = (0, 1), while + + =, + + =, + + = has an infinite number of solutions because the third equation is the first equation plus twice the second one and hence contains no independent information; thus any value of z can be chosen and values of x and y can be ...
Because consistency of ZF is not provable in ZF, the weaker notion relative consistency is interesting in set theory (and in other sufficiently expressive axiomatic systems). If T is a theory and A is an additional axiom , T + A is said to be consistent relative to T (or simply that A is consistent with T ) if it can be proved that if T is ...
In database systems, consistency (or correctness) refers to the requirement that any given database transaction must change affected data only in allowed ways. Any data written to the database must be valid according to all defined rules, including constraints , cascades , triggers , and any combination thereof.
Configuration Identification (CI): consists of setting and maintaining baselines, which define the system or subsystem architecture, components, and any developments at any point in time. It is the basis by which changes to any part of a system are identified, documented, and later tracked through design, development, testing, and final delivery.
Consistency models are used in distributed systems like distributed shared memory systems or distributed data stores (such as filesystems, databases, optimistic replication systems or web caching). Consistency is different from coherence, which occurs in systems that are cached or cache-less, and is consistency of data with respect to all ...
An axiomatic system is said to be consistent if it lacks contradiction.That is, it is impossible to derive both a statement and its negation from the system's axioms. Consistency is a key requirement for most axiomatic systems, as the presence of contradiction would allow any statement to be proven (principle of explo
The ω-consistency of a system implies its consistency, but consistency does not imply ω-consistency. J. Barkley Rosser strengthened the incompleteness theorem by finding a variation of the proof (Rosser's trick) that only requires the system to be consistent, rather than ω-consistent. This is mostly of technical interest, because all true ...
If an underdetermined system of t equations in n variables (t < n) has solutions, then the set of all complex solutions is an algebraic set of dimension at least n - t. If the underdetermined system is chosen at random the dimension is equal to n - t with probability one.