The collection of all relations on a set.

Question

I have the following question for homework

I'm unsure what the first part of the question refers to. I'm assuming its asking for the size of the power set of A but I haven't seen any mention of power sets regarding relations.

The second part refers to the size of the Cartesian product A^2(A * A).

The third and fourth part I don't recall seeing in my text book and I can't seem to find any mention of "collection of relations" online.

score 1 · Answer 1 · answered Sep 19 '17 at 03:34

1

$C$ is not mentioned in the first part. You are correct that it is just looking for the size of the power set of $A$ and $8$ is correct. The third part is asking you to find how many different relations there are on $A$. You should review the definition of a relation to find the size of this set. Since a relation is just a set of ordered pairs, one can certainly talk of the Cartesian product of two of these sets.

answered Sep 19 '17 at 03:34

Ross Millikan

374,822

Thanks for the explanation. I didn't realize that just being an ordered pair is considered a relation. I assumed that there had to be some kind of relationship between the ordered pairs for the relation to exist. – stalris Sep 19 '17 at 04:22
A relation is a set of ordered pairs from the underlying set. Just like functions, there doesn't have to be any easily stated rule for which pairs are in the relation and which are not. Very few relations will have a nice rule. That should be your definition of a relation-read definitions carefully. – Ross Millikan Sep 19 '17 at 04:27

score 1 · Accepted Answer · answered Sep 19 '17 at 03:34

Recall a relation on a set $A$ is a subset of $A \times A$. Note that since $|A|=3$ then $|A \times A| = 9$. Letting $C$ be the collection relations on $A$, we then would have:

\begin{array} \ |C| &= \text{ amount of subsets of $A \times A$ } \\ &= | \mathcal{P}(A\times A)| \\ \end{array}

The collection of all relations on a set.

2 Answers2