Content deleted Content added
No edit summary |
|||
Line 271:
of the two theories is identical: the same abstraction is implemented by these two superficially
different approaches.
== Equivalence relations and partitions ==
A general technique for implementing abstractions in set theory is the use of equivalence
classes. If an equivalence relation R tells us that elements of its field A are alike in
some particular respect, then for any set x we can regard the set <math>[x]_R</math> as
representing an abstraction from the set x respecting just those features (we identify
elements of A up to R).
For any set A, we say that a set <math>P</math> is a <strong>partition</strong> of A if all elements of P
are nonempty, any two distinct elements of P are disjoint, and <math>A=\bigcup P</math>.
For every equivalence relation R with field A, <math>\{[x]_R \mid x \in A\}</math>
is a partition of A. Moreover, each partition P of A determines an equivalence relation
<math>\{(x,y) \mid (\exists A \in P.x \in A \wedge y \in A)\}</math>.
| |||