Question:
Prove that the intersection of two compact sets is compact, using criterion (2).
Prove that the intersection of two compact sets is compact, using criterion (1).
Prove that the intersection of two compact sets is compact, using criterion (3).
Probably the most important new idea you'll encounter in real analysis is
the concept of compactness. It's the compactness of [a, b] that makes a
continuous function reach its maximum and that makes the Riemann in-
tegral exist. For...
Using the definition of Compact Set, prove that the union of two
compact sets is compact. Use this result to show that the union of
a finite collection of compact sets is compact. Is the union of any
collection of compact sets compact?
1)Prove that the intersection of an arbitrary collection of
closed sets is closed.
2)Prove that the union of a finite collection of closed sets is
closed
Prove that a disjoint union of any finite set and any countably
infinite set is countably infinite.
Proof: Suppose A is any finite set,
B is any countably infinite set, and A and
B are disjoint. By definition of disjoint, A ∩ B = ∅
In case A = ∅, then A ∪ B = B, which
is countably infinite by hypothesis.
Now suppose A ≠ ∅. Then there is a positive integer
m so that A has m elements...
Unless otherwise noted, all sets in this module are finite.
Prove the following statements...
1. If A and B are sets then (a) |A ∪ B| = |A| + |B| − |A ∩ B|
and (b) |A × B| = |A||B|.
2. If the function f : A→B is (a) injective then |A| ≤ |B|. (b)
surjective then |A| ≥ |B|.
3. For each part below, there is a function f : R→R that is (a)
injective and surjective. (b)...