# Greatest Hits on HW 7

1. (4.3) 25 was often skipped/done wrong. Will do in class on 4/11.
2. A binary relation on A is a different object from a function from A to A. This occurred a lot in the proof-based questions.

3. Please review the definition of the composition of two relations.
4. Suppose R\subseteq A\times B and S\subseteq B\times C are two relations.

• Then R\circ S=\{(a,c) : a\in A \land c\in C \land \exists b\in B, (a,b)\in R \land (b,c) \in S\}.

• Graphically, R\circ S consists of all endpoints in A\times C that can be connected via B using arrows in the relations R and S.

• If R\subseteq A\times A is composed with itself: R\circ R consists of all endpoints of "paths of length 2" that can be formed in the arrow diagram.

5. Not every set has an order relation ('<') defined on it (i.e. be sure not to confuse an ordered set like the natural numbers and any set in general).

6. When trying to come up with examples, try to keep it simple. E.g. the simplest example of a transitive and symmetric relation is the empty relation.
7. The grader recommended you guys don't use the proof techniques on this page. He said proof by illegibility and proof mumbo-jumbo were especially popular.