3
votes
5answers
295 views

Real life examples of order relations.

It's easy to find examples of equivalence relations (for example, A shares room with B), but I can't seem to find a real life example of an order relation (that is, a relation that's reflexive, ...
0
votes
1answer
15 views

What other statements of this general form can we prove about the direct image function?

Given a relation $R : X \rightarrow Y,$ write $R_* : \mathcal{P}(X) \rightarrow \mathcal{P}(Y)$ for the direct image function defined by asserting that $$b \in R_*(A) \Leftrightarrow \exists a \in A : ...
1
vote
0answers
120 views

Example relations: pairwise versus mutual

There are by now several questions on math.se asking about pairwise versus mutual relations, eg: • When does “pairwise” strengthen and when does it weaken? • Relation: pairwise and mutually • ...
2
votes
0answers
199 views

Examples of proofs by induction with respect to relations that are not strict total orders.

I have read this Wikipedia article and found it fascinating. I came across it after I tried to prove a certain statement with a method resembling induction in the set of natural numbers but ordered by ...