a.
Prove that an infinite subset of a denumerable set is denumerable.
a.
Answer to Problem 9E
Proved
Explanation of Solution
Given information:
Use the theorems of this section.
Calculation:
We have to use the theorem to prove an infinite subset of a denumerable set is denumerable.
A denumerable set
Every subset of a countable set is countable so
Since
Hence, an infinite subset of a denumerable set is denumerable.
b.
Prove that if
b.
Answer to Problem 9E
Proved
Explanation of Solution
Given information:
Use the theorems of this section.
Calculation:
We have to use the theorem to prove
Consider an uncountable set
It implies that set
The assumption
Hence, if
c.
Prove that
c.
Answer to Problem 9E
Proved
Explanation of Solution
Given information:
Use the theorems of this section.
Calculation:
We have to use the theorem to prove
The set
Hence,
d.
Prove that
d.
Answer to Problem 9E
Proved
Explanation of Solution
Given information:
Use the theorems of this section.
Calculation:
We have to use the theorem to prove
The set
The set
Since the set
Hence,
The set
Hence,
e.
Prove that
e.
Answer to Problem 9E
Proved
Explanation of Solution
Given information:
Use the theorems of this section.
Calculation:
We have to use the theorem to prove
The set
Hence,
f.
Prove that
f.
Answer to Problem 9E
Proved
Explanation of Solution
Given information:
Use the theorems of this section.
Calculation:
We have to use the theorem to prove
The set
Since the set of natural numbers is denumerable. the countable collection of countable sets is countable.
Hence,
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