Let’s find the equivalence classes of the following finite set S:
Given we can form the following relation
Note: writing the relation on set in the following ways is equivalent:
This relation, has been given the symbol but it means “the same sign and parity” in this case. For instance, or tells us that one and three are both odd and both have the same sign in set (both positive).
The equivalence classes for this relation are the following sets:
We obtained the above equivalence classes by asking ourselves:
- How is the element related to any other element in the set under the definition of
Since R is defined as “the same sign and same parity,” then we’re really asking ourselves whether has the same sign as any other element in Since all the other elements are positive, then has the equivalence class containing only itself. Another question we would’ve asked ourselves is whether is even or odd. …