Session 7C
Partitions
45-75 min - work through lesson notes, practice, and the MCQ check.
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Session tips
- Finish the MCQ before marking complete.
- Check the answer outlines after trying on paper.
- Mark complete to update your certificate progress.
Lesson notes
- A partition is a collection of nonempty subsets that are pairwise disjoint and whose union is the entire set. Equivalence relations automatically create partitions: the equivalence classes form the blocks of the partition. Two key facts make this work. First, classes are either identical or disjoint: if two classes share an element, then by transitivity they must be the same class. Second, every element lies in its own equivalence class, so the union of all classes is the whole set.
- These are proof exercises that reinforce controlled logical reasoning. To show "classes are disjoint or equal," you assume two classes overlap, pick an element in the overlap, and use the relation properties to show every element of one class must relate to every element of the other in the right way, forcing equality. This style is similar to set equality proofs, but now the structure of the relation (especially transitivity) does the heavy lifting.
Practice
- 1Prove equivalence classes are either disjoint or identical.
- 2Prove union of all classes is the whole set.
MCQ
Two equivalence classes are: