Reading Quiz Section 6.3
1. Let I be a set and {A
n
: n ∈ I} a family of sets indexed by I. The definition of
S
n∈I
A
n
uses the
quantifier and the definition of
T
n∈I
A
n
uses the quantifier.
(a) existential; existential
(b) existential; universal
(c) universal; existential
(d) universal; universal
2. Let {A
n
: n ∈ I} be a nested collection of sets where A
1
⊇ A
2
⊇ A
3
⊇ · · · . What can you
conclude? Select all that apply.
(a)
T
n∈I
A
n
= ∅.
(b)
S
n∈I
A
n
= A
1
.
(c) The collection of sets is pairwise disjoint.
(d) Each A
n
must be an interval.
3. True or False:
B ⊆
[
n∈I
A
n
⇐⇒ ∀n ∈ I, B ⊆ A
n
Practice Problems Section 6.3
1. For each non-negative real number r ≥ 0 let
A
r
=
(x, y) ∈ R
2
: x
2
+ y
2
= r
2
(a) Describe each of the sets A
r
geometrically.
(b) Prove that
S
r∈R
+
0
A
r
= R
2
.
Video Solution
2. Let {A
n
: n ∈ I} be an indexed collection of sets and B a set. Prove:
(a)
[
n∈I
A
n
!
∩ B =
[
n∈I
(A
n
∩ B)
(b)
\
n∈I
A
n
!
∪ B =
\
n∈I
(A
n
∪ B)
Video Solution
