Sets, relations, and equivalence relations

Published

2023-08-05

Sets: a quick review

Relations, equivalence relations, and equivalence classes

When we speak of “equivalence relations” we’re not speaking of equivalent as being identical or the same object. We are speaking of equivalence with respect to a particular relation. This is common enough in colloquial usage: a chickadee and a hawk are both birds, a Toyota Corolla and a Subaru Crosstrek are both cars—so in some sense these are “equivalent.” In the context of equivalence relations, we give the common example of congruence modulo n. In the case of congruence modulo 5, 1 and 6 are equivalent. The both have the same remainder (also called the residue) and thus they belong to the same equivalence class.

Comprehension check:

  1. True or false? The union of two sets, A and B, is the set containing all elements of A and all the elements of B.
  2. True or false? The intersection of two sets, A and B, is the set containing only those elements which are in A and in B.
  3. True or false? The empty set isn’t a subset of any set.
  4. True or false? Any set is a subset of itself.
  5. The Cartesian product A \times B of A = \{1, 2, 3\} and B = \{Fred, Ethel \} is ________________________________.
  6. We may think of a function as a set of ordered ____________, which is a subset of the __________________ of its domain and its codomain.
  7. Equivalence classes form a ________________ of the underlying set.
  8. While friendship is a relation, it is not an equivalence relation because it violates the requirement of __________________.
  9. Given some equivalence relation R and equivalence class q where q is a representative of the class, then the set of all elements x in q are those elements for which ________________ is true.

Answers: b ɹ x / ʎʇᴉʌᴉʇᴉsuɐɹʇ / uoᴉʇᴉʇɹɐd / ʇɔnpoɹd uɐᴉsǝʇɹɐƆ / sɹᴉɐd / {(lǝɥʇƎ ‘Ɛ)’(lǝɥʇƎ ‘ᄅ)’(lǝɥʇƎ ‘Ɩ)’(pǝɹℲ ‘Ɛ)’(pǝɹℲ ‘ᄅ)’(pǝɹℲ ’Ɩ)} / ǝnɹʇ / ǝslɐɟ / ǝnɹʇ / ǝnɹʇ

Original author: Clayton Cafiero < [given name] DOT [surname] AT uvm DOT edu >

No generative AI was used in producing this material. This was written the old-fashioned way.

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