Notes for 8 September

- 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.

(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.

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.

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.

- 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.

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 Microsoft^{TM} 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.