Chapter 13
Notes for 8 September

Notes also available as PDF.

13.1 Review

An unordered collection of unique elements.
Any item in a set, even other sets. (Also entry, member, item, etc.)
empty set
Or null set. Denoted by rather than \{\}.
A set with only one element.

13.2 Relations and Venn diagrams

(Someday I will include Venn diagrams for these in the notes.)

element of
The expression x ∈ A states that x is an element of A. If x\mathrel{∉}A, then x is not an element of A.
If all entries of set A also are in set B, A is a subset of B.
The reverse of subset. If all entries of set B also are in set A, then A is a superset of B.
proper subset
If all entries of set A also are in set B, but some entries of B are not in A, then A is a proper subset of B.
Set A equals set B when A is a subset of B and B is a subset of A.

The symbols for these relations are subject to a little disagreement.

13.3 Translating relations into (and from) English

From English:

To English:

13.4 Consequences of the set relation definitions

Every set is a subset of itself. Expected.

If A = B, then every member of A is a member of B, and every member of B is a member of A. This is what we expect from equality, but we did not define set equality this way. Follow the rules:

The empty set is a subset of all sets. Unexpected! This is a case of carrying the formal logic to its only consistent end.

The alternatives would not be consistent, but proving that requires more machinery that we need.

13.5 Operations

The union of two sets A and B, denoted by A ∪ B, is the set consisting of all elements from A and B.
The intersection of two sets A and B, denoted A ∩ B, is the set consisting of all elements that are in both A and B.
set difference
The set difference of two sets A and B, written A \ B, is the set of entries of A that are not entries of B.

If A and B share no entries, they are called disjoint. One surprising consequence is that every set A has a subset disjoint to the set A itself.

Can any other subset be disjoint with its superset? No.

13.6 Homework

Groups are fine, turn in your own work. Homework is due in or before class on Mondays.

Write out (briefly) your approach to each problem.

Note that you may email homework. However, I don’t use MicrosoftTM products (e.g. Word), and software packages are notoriously finicky about translating mathematics.

If you’re typing it (which I advise just for practice in whatever tools you use), you likely want to turn in a printout. If you do want to email your submission, please produce a PDF or PostScript document.