Equivalence Relations Discussion

(The following is/are the initial prompt(s) for an online discussion; students may have posted responses, and prompts for further discussion may have been added, but these things are not shown.)

Equivalence relations are ones that behave in certain ways like the familiar equality relation. In particular, equivalence relations are reflexive (every value is equivalent to itself), symmetric (if a is equivalent to b, then b is also equivalent to a), and transitive (if a is equivalent to b and b to c, then a is equivalent to c). Many useful relations in mathematics turn out to be equivalence relations, so it is helpful to be able to recognize them and to understand their properties. This discussion starts building those abilities.

Consider the people in some historical or fictional group (e.g., the  Tudor family tree from class, or the characters in some novel, film, TV series, etc.) Can you find an equivalence relation on that group? Can you find relations on that group that aren’t quite equivalence relations, in that they have some but not all of the necessary properties? What properties do these relations have, and what ones are they missing?