WEEK
3
M: Inclusion and basic operations on sets.
2.3. Definition.
Inclusion. Subsets. Proper subsets.
2.4. Proposition.
Basic properties of inclusion. (a) Ø is a subset of any set,
(b) reflexivity, (c) transitivity,
(d) antisymmetricity --
extensionality.
2.5. Definition.
Basic operations on sets. (a) intersection, (b) union, (c) set difference,
(d) symmetric difference,
(e) complement ,
2.6. Proposition.
Basic facts about set operations. (a) intersection and union with Ø,
(b) commutativity,
(c) associativity, (d) distributivity,
(e) De Morgan Laws -- with proof of one of them.
PROBLEMS FOR DISCUSSIONS:
Book, p.44 Exercise 3, 10, 13, 14, 17 and p.52 Exercise
22, 23 , 24, 30
W: Power set. General unions and intersections. Examples.
2.7. Definition. Power
set.
Examples: Power set of
one, two element sets, P(Ø), P(P(Ø)), P(N).
2.8. Definition. General
union and intersection.
Examples: Union of two
sets, union of finitely many sets. union of all Ap={n \in N+
| n divisible by p},
union of all intervals In
= (n, n+1).
PROBLEMS FOR DISCUSSIONS:
Book, p.44 Exercise 11, 12, 18, 19, 20 and p.53 Exercise
25, 34, 35, 43, 48
F: More examples of infinite unions and intersections. Completion
of the example with In from Wed. Viewing
the collections of {Ap| p prime} and {In
| n integer}as examples of indexed systems.
Examples: An open square can be expressed
as the union of a set of open disks.
A closed square cen be expressed as the intersection of a set
of open disks. (Without proof.)
PROBLEMS FOR DISCUSSIONS:
Book, p.54 Exercise 49, 52 and p.59 Exercise 54, 55, 57b, 58, 71