Kenneth Rosen Edition 7 Exercise 2.5 Question 1 (Page No. 176)

Question

Determine whether each of these sets is finite, countably infinite, or uncountable. For those that are countably infinite, exhibit a one-to-one correspondence between the set of positive integers and that set.

• A. the negative integers
• B.  the even integers
• C. the integers less than $100$
• D. the real numbers between $0$ and $\frac{1}{2}$
• E. the positive integers less than $1,000,000,000$
• F. the integers that are multiples of $7$

Answer 1

Theorem: Every subset of a countable set is countable. In particular, every infinite subset of a countably infinite set is countably infinite.

A) countably infinite as we have one-to-one correspondence between positive integers and negative integers.

B) countably infinite as it is subset of Natural numbers

C) countably infinite as one-to-one correspondence can be formed by $f(n)= 100-n$

D) uncountable as it is not possible to list all real numbers between $0$ and $\frac{1}{2}$

E) finite

F) countably infinite as it is subset of Natural numbers