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 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...
Cardinality
State whether the following sets are finite, countable infinite
or uncountable.
Set of positive perfect squares.
Is it finite, countable infinite or uncountable? If it is
countably infinite, set up the bijection between
ℤ+.
Negative numbers greater than or equal to -5.
Is it finite, countable infinite or uncountable? If it is
countably infinite, set up the bijection between
ℤ+.
Odd positive integers.
Is it finite, countable infinite or uncountable? If it is
countably infinite, set up the bijection...
1)Show that a subset of a countable set is also countable.
2) Let P(n) be the statement that 13 + 23 +· · ·+n3 =(n(n +
1)/2)2 for the positive integer n.
a) What is the statement P(1)?
b) Show that P(1) is true, completing the basis step of
the proof.
c) What is the inductive hypothesis?
d) What do you need to prove in the inductive step?
e) Complete the inductive step, identifying where you
use the inductive hypothesis....
2. (a) Prove that U45 is generated by the set {14,28}.
(b) Prove that the additive group Z×Z is generated by the set
S={(3,1),(−2,−1),(4,3)}.
Please be thorough step by step with details, please.