Kenneth Rosen Edition 7 Exercise 2.5 Question 2 (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 integers greater than $10$
• B. the odd negative integers
• C. the integers with absolute value less than $1,000,000$
• D. the real numbers between $0$ and $2$
• E. the set $A \times Z^{+}$ where $A = \{2, 3\}$
• F. the integers that are multiples of $10$

Answer 1

A set is countably infinite if its elements can be put in one-to-one correspondence with the set of natural numbers i.e. $f:\mathbb{N}\rightarrow A$

A) countably infinite as one-to-one correspondence can be given by $f(n)=n+10$

B) countably infinite as one-to-one correspondence can be given by $f(n)=-(2n-1)$

C) finite as set contains {${-999999, -999998,.........,-1,0,1,..........999998,999999}$}

D) uncountable, not possible to list all real numbers between $0$ and $2$

E) countably infinite as one-to-one correspondence can be given by $f(n)=\left\{\begin{matrix} (2,n/2) &if\: n\: is\: even \\ (3,(n+1)/2)& if\: n\: is\: odd \end{matrix}\right.$

F) countably infinite as one-to-one correspondence can be given by $f(n)=\left\{\begin{matrix} 0 & if \: n=1\\ 5n & if\: n\: is\: even \\ -5(n-1) & if\: n\: is\: odd\: and\: n\neq 1 \end{matrix}\right.$