Question

Prove that S={1,1/4,1/8,1/16,...16,...} is countable

Answer #1

Determine the Huffman encoding of a source with probabilities
1/3,1/4,1/6,1/8,1/12,1/24. Compare the average word length to the
entropy of the source.

Prove whether or not the set ? is countable.
a. ? = [0, 0.001)
b. ? = ℚ x ℚ
I do not really understand how to prove
S is countable.

Use the fact that “countable union of disjoint countable sets is
countable" to prove “the set of all polynomials with rational
coefficients must be countable.”

Prove the union of a finite collection of countable
sets is countable.

Prove that a countable union of countable sets countable; i.e.,
if {Ai}i∈I is a collection of sets, indexed by I ⊂ N, with each Ai
countable, then union i∈I Ai is countable. Hints: (i) Show that it
suffices to prove this for the case in which I = N and, for every i
∈ N, the set Ai is nonempty. (ii) In the case above, a result
proven in class shows that for each i ∈ N there is a...

Prove for each:
a. Proposition: If A is finite and B is countable, then A ∪ B is
countable.
b. Proposition: Every subset A ⊆ N is finite or countable.
[Similarly if A ⊆ B with B countable.]
c. Proposition: If N → A is a surjection, then A is finite or
countable. [Or if countable B → A surjection.]

Show that the relation
R={(1,1),(1,4),(2,2),(2,3),(3,3),(3,2),(4,1),(4,4)} is an
equivalence relation and contrust the associated directed
graph.

prove that the lebesgue measure on R has a countable
basis

[Q] Prove or disprove:
a)every subset of an uncountable set is countable.
b)every subset of a countable set is countable.
c)every superset of a countable set is countable.

Find the interior and closure of each set .
(a) [0,∞)
(b) (1,1/2)∪(1/2,1/3)∪(1/3,1/4)∪(1/4,1/5)∪...
(c){1 + 1/2 + 1/3 +···+ 1/n:n∈N}
(d){1 + 1/4 + 1/16 +···+ 1/4^n:n∈N}

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