Activity 18.1.
For each of the relations provided, carry out the following steps.
- Verify that the relation is an equivalence relation on the set
- Consider a few example equivalence classes, for the specific example representative elements provided (if applicable). What other elements are in that class?
- Devise a general way to describe every equivalence class, using your experience from the example classes already considered (if applicable). Make your class descriptions more meaningful than just “all elements equivalent to a specific representative element.”
- List/describe all elements in the quotient
(a)
Relation on where means Example equivalence classes for
(b)
Relation on where means Example equivalence classes for
(c)
Relation on where means Example equivalence classes for
(d)
Relation on where means Example equivalence classes for
(e)
Relation on the vertex set of a graph where means there exists a path in from to
(f)
Given function the relation on the domain where means