Let Σ an and Σ bn be series with positive
terms such that lim(an/bn) =
λ ∈ (0, ∞). Prove that the two series have the same behavior, that
is they both converge or they both diverge to +∞.
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) injective but not surjective.
(c) surjective but not injective.
(d)...
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. Let f : A→B and g : B→C be surjective....
(a) Let Λ = {λ ∈ R : 0 < λ < 1}. For each λ ∈ Λ, let Aλ =
{x ∈ R : −λ < x < 1/λ}. Find U λ∈Λ Aλ and ~U λ∈Λ Aλ
respectively.
(b) Let Λ = \ {λ ∈ R : λ > 1}. For each λ ∈ Λ, let Aλ = {x ∈
R : −λ < x < 1/λ}. Find U λ∈Λ Aλ and ~U λ∈Λ Aλ
respectively.
Prove the following statements!
1. There is a bijection from the positive odd numbers to the
integers divisible by 3.
2. There is an injection f : Q→N.
3. If f : N→R is a function, then it is not surjective.