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...
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
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)...
Unless otherwise noted, all sets in this module are finite.
Prove the following statements...
1. Let S = {0, 1, . . . , 23} and define f : Z→S by f(k) = r
when 24|(k−r). If g : S→S is defined by (a) g(m) = f(7m) then g is
injective and (b) g(m) = f(15m) then g is not injective.
2. Let f : A→B and g : B→C be injective. Then g ◦f : A→C is
injective.
3....