# Collections of Sets, Problem 3

### Hints for (a)

$$\sim$$ is a relation on $$S=\mathbb{P}(\mathbb{Z})$$. That means that each element of the base set S is a subset of the integers. So $$\sim$$ compares one subset of the integers (A) to another subset of the integers (B).

Try setting A and B to specific familiar sets. For example, set them both to finite sets. What is their symmetric difference? Does the relation hold?

Now, repeat this with A and B set to some familiar infinite sets of integers. Again, what is the symmetric difference and are they related by $$\sim$$?

### Hints for (b)

Write out the start and end of a standard subset inclusion proof. E.g. pick an element from (A-C) and then show that it is in $$(A-B) \cup (B - C)$$.

The middle of the proof should work with statements about inclusion in individual sets. E.g. x is an element of B and x is not an element of A.

It may help to draw a Venn diagram to understand why the claim is true.

### Hints for (c)

Reflexive and symmetric shouldn't be hard. To attack transitive, think about using your result from part (b).

Suppose that $$(A - B) \cup (B - A)$$ is finite. What does this imply about (A-B) and (B-A)?