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)?