Use induction to prove that the union of n countable sets is
countable where n is...
Use induction to prove that the union of n countable sets is
countable where n is a positive integer. (can use the fact that
union of two countable sets is countable)
(11) Prove that a union of two countable sets is countable.
(Hint: the same idea used to show that Z is countable might be
useful.) (Don’t forget that countable sets can be finite.)
(12) We saw in class that N × N ∼ N is countable. Prove that A ×
B is is countable for any countable sets A, B. (Hint: If you can
prove that A × B ∼ N × N then you can use what has already...
1. Use induction to prove that Summation with n terms where i=1
and Summation 3i 2 − 3i + 1 = n^3 for all n ≥ 1.
2. Let X be the set of all natural numbers x with the property
that x = 4a + 13b for some natural numbers a and b. For example, 30
∈ X since 30 = 4(1) + 13(2), but 5 ∈/ X since there’s no way to add
4’s and 13’s together to...
4: \textbf{Proof} Prove that if $A$ and $B$ are countable sets,
then $A \cup B$ is countable.
5: Use induction and problem 4 to prove that if $A_1, A_2, ...,
A_m$ are each countable sets, then the union $A_1 \cup A_2 \cup ...
\cup A_m$ is countable.
#5 please
Use induction to prove
Let f(x) be a polynomial of degree n in Pn(R). Prove that for
any g(x)∈Pn(R) there exist scalars c0, c1, ...., cn such that
g(x)=c0f(x)+c1f′(x)+c2f′′(x)+⋯+cnf(n)(x), where f(n)(x)denotes the
nth derivative of f(x).