(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. Myuse of \mathbf{⊂}is for subsets and not proper subsets.
If A = B, then everymember of Ais amember of B, andevery member of Bis 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.
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.