“equivalence relation”的概念、定义、翻译、参考文献-科学参考

equivalence relation

Mathematics

A binary relation ∼ on a set S that is reflexive, symmetric and transitive. For an equivalence relation ∼, a is said to be equivalent to b when a ∼ b. It is an important fact that, from an equivalence relation on S, equivalence classes can be defined to obtain a partition of S. Conversely, given any partition of S, an equivalence relation can be defined by a ∼ b if and only if a and b are in the same part of the partition.

Computer

A relation that is transitive, symmetric, and reflexive. The concept is a convenient generalization or abstraction of equality. It covers most notions of equals, equivalence, and similarity as defined between triangles, algorithms, Boolean expressions, algebraic structures, statements, etc. See also equivalence class, partial ordering.

Philosophy

A relation which is transitive, symmetric, and reflexive divides its field into exclusive classes of things. Within each class everything bears the relation to everything else, and nothing bears it to anything in a different class. ‘Of the same height as’ or ‘having the same number of members as’ are equivalence relations that divide the field of people into classes of equally tall people, and sets into different classes depending on how many members they have. Defining an equivalence class of things at the same point on the scale is the fundamental operation in establishing any kind of measure of a property or quantity. In mathematical logic an early definition of a cardinal number (Frege’s) was as an equivalence class of equinumerous classes.

单词	equivalence relation
释义	equivalence relation Mathematics A binary relation ∼ on a set S that is reflexive, symmetric and transitive. For an equivalence relation ∼, a is said to be equivalent to b when a ∼ b. It is an important fact that, from an equivalence relation on S, equivalence classes can be defined to obtain a partition of S. Conversely, given any partition of S, an equivalence relation can be defined by a ∼ b if and only if a and b are in the same part of the partition. Computer A relation that is transitive, symmetric, and reflexive. The concept is a convenient generalization or abstraction of equality. It covers most notions of equals, equivalence, and similarity as defined between triangles, algorithms, Boolean expressions, algebraic structures, statements, etc. See also equivalence class, partial ordering. Philosophy A relation which is transitive, symmetric, and reflexive divides its field into exclusive classes of things. Within each class everything bears the relation to everything else, and nothing bears it to anything in a different class. ‘Of the same height as’ or ‘having the same number of members as’ are equivalence relations that divide the field of people into classes of equally tall people, and sets into different classes depending on how many members they have. Defining an equivalence class of things at the same point on the scale is the fundamental operation in establishing any kind of measure of a property or quantity. In mathematical logic an early definition of a cardinal number (Frege’s) was as an equivalence class of equinumerous classes.
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