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The term database is correctly applied to the data and their supporting data structures, and not to the database management system (DBMS). The database data collection with DBMS is called a database system.
The term database system implies that the data are managed to some level of quality (measured in terms of accuracy, availability, usability, and resilience) and this in turn often implies the use of a general-purpose database management system (DBMS). A general-purpose DBMS is typically a complex software system that meets many usage requirements to properly maintain its databases which are often large and complex.
This is specially the case with client-server, near-real time transactional systems, in which multiple users have access to data, data is concurrently entered and inquired for in ways that preclude single-thread batch processing. Most of the complexity of those requirements are still present with personal, desktop-based database systems.
The utilization of databases is now so widespread that virtually every technology and product relies on databases and DBMSs for its development and commercialization, or even may have DBMS software embedded in it. Also, organizations and companies, from small to large, depend heavily on databases for their operations.
Well known DBMSs include FoxPro, IBM DB2, Linter, Microsoft Access, Microsoft SQL Server, MySQL, Oracle, PostgreSQL and SQLite. A database is not generally portable across different DBMS, but different DBMSs can inter-operate to some degree by using standards like SQL and ODBC together to support a single application built over more than one database. A DBMS also needs to provide effective run-time execution to properly support (e.g., in terms of performance, availability, and security) as many database end-users as needed.