Let '~' denote an equivalence relation over some nonempty set A, called the universe or underlying set. Let G denote the set of bijective functions over A that preserve the partition structure of A, meaning that for all and Then the following three connected theorems hold:Rosen (2008), pp. 243–45. Less clear is §10.3 of Bas van Fraassen, 1989. Laws and Symmetry. Oxford Univ. Press. ~ partitions A into equivalence classes. (This is the , mentioned above); Given a partition of A, G is a transformation group under composition, whose orbits are the cells of the partition; Given a transformation group G over A, there exists an equivalence relation ~ over A, whose equivalence classes are the orbits of G.Wallace, D. A. R., 1998. Groups, Rings and Fields. Springer-Verlag: 202, Th. 6.Dummit, D. S., and Foote, R. M., 2004. Abstract Algebra, 3rd ed. John Wiley & Sons: 114, Prop. 2.
Let '~' denote an equivalence relation over some nonempty set A, called the universe or underlying set. Let G deno?te the set of bijective functions over A that preserve the partition structure of A, meaning that for all and Then the following three connected theorems hold:Rosen (2008), pp. 243–45. Less clear is §10.3 of Bas van Fraassen, 1989. Laws and Symmetry. Oxford Univ. Press. ~ partitions A into equivalence classes. (This is the , mentioned above); Given a partition of A, G is a transformation group under composition, whose orbits are the cells of the partition; Given a transformation group G over A, there exists an equivalence relation ~ over A, whose equivalence classes are the orbits of G.Wallace, D. A. R., 1998. Groups, Rings and Fields. Springer-Verlag: 202, Th. 6.Dummit, D. S., and Foote, R. M., 2004. Abstract Algebra, 3rd ed. John Wiley & Sons: 114, Prop. 2.