## Chapter 13Notes for 8 September

Notes also available as PDF.

### 13.1 Review

set
An unordered collection of unique elements.
• Curly braces: \{A,B,C\} is a set of three elements, A, B, and C.
• Can be implicit or in set builder notation: \{x\kern 1.66702pt |\kern 1.66702pt x\text{ is an integer},x > 0,x < 3\} is the same set as \{1,2\}.
• Order does not matter, repeated elements do not matter.
element
Any item in a set, even other sets. (Also entry, member, item, etc.)
empty set
Or null set. Denoted by rather than \{\}.
• This is a set on its own.
• \{∅\} is the set of the empty set, which is not empty.
singleton
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.
• 4 \{2,4,6\}, and 4\mathrel{∉}\{x\kern 1.66702pt |\kern 1.66702pt x\text{ is an odd integer }\}.
• There is no x such that x , so \{x\kern 1.66702pt |\kern 1.66702pt x ∅\} is a long way of writing .
subset
If all entries of set A also are in set B, A is a subset of B.
superset
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.
• \{2,3\} is a proper subset of \{1,2,3,4\}.
equality
Set A equals set B when A is a subset of B and B is a subset of A.
• Order does not matter. \{1,2,3\} = \{3,2,1\}.

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

• Many basic textbooks write the subset relation as , so A ⊆ B when A is a subset of B. The same textbooks reserve for the proper subset. Supersets are .
• This keeps a superficial similarity to the numerical relations and <. In the former the compared quantities may be equal, while in the latter they must be different.
• Most mathematicians now use for any subset. If a property requires a “proper subset”, it often is worth noting specifically. And the only non-“proper subset” of a set is the set itself.
• Extra relations are given for emphasis, e.g. or for proper subsets and or to emphasize the possibility of equality.
• Often a proper subset is written out: A ⊂ B and A≠B.
• I’ll never remember to stick with the textbook’s notation. My use of \mathbf{⊂} is for subsets and not proper subsets.

### 13.3 Translating relations into (and from) English

From English:

• The train has a caboose.
• It’s reasonable to think of a train as a set of cars (they can be reordered).
• The cars are the members.
• Hence, caboose train
• The VI volleyball team consists of VI students.
• VI volleyball team VI students
• There are no pink elephants.
• pink elephants = ∅

To English:

• x today’s homework set.
• x is a problem in today’s homework set.
• Today’s homework this week’s homework.
• Today’s homework is a subset of this week’s homework.

### 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:

• A = B imples A ⊂ B and B ⊂ A.
• Because A ⊂ B, every member of A is a member of B.
• Because B ⊂ A, every member of B is a member of A.

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

• For some set A, ∅⊂ A if every member of is in A.
• But has no members.
• Thus all of ’s members also are in A.
• This is called a vacuous truth.

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

### 13.5 Operations

union
The union of two sets A and B, denoted by A ∪ B, is the set consisting of all elements from A and B.
• A ∪ B = \{x\kern 1.66702pt |\kern 1.66702pt x A\text{ or }x B\}.
• Remember repeated elements do not matter: \{1,2\} ∪\{2,3\} = \{1,2,3\}.
intersection
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.
• A ∩ B = \{x\kern 1.66702pt |\kern 1.66702pt x A\text{ and }x B\}.
• \{1,2\} ∩\{2,3\} = \{2\}.
• \{1,2\} ∩\{3,4\} = \{\} = ∅.
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.
• A \ B = \{x\kern 1.66702pt |\kern 1.66702pt x A\text{ and }x∉B\}.
• Sometimes written as A - B, but that often becomes confusing.

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.

• No sets (not even ) can share elements with because has no elements.
• So all sets are disjoint with .
• The empty set is a subset of all sets.
• So all sets are disjoint with at least one of their subsets!

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.

• Problem set 2.1 (p83):
• Problems 7, 8, 10, 20, 24

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.