Determine whether each of these sets is finite, countably infinite, or uncountab
ID: 3127181 • Letter: D
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
___The set is countably infinite.
___The set is finite.
___The set is countably infinite with one-to-one correspondence 1 11, 2 12, 3 13, and so on.
___The one-to-one correspondence is given by n (n + 10).
___The set is countably infinite with one-to-one correspondence 1 10, 2 12, 3 17, and so on.
___The one-to-one correspondence is given by n (2n + 10).
b) The odd negative integers
___The set is countably infinite.
___The set is finite.
___The set is countably infinite with one-to-one correspondence 1 –1, 2 –3, 3 –5, 4 –7, and so on.
___The set is countably infinite with one-to-one correspondence 1 –1, 2 –2, 3 –3, 4 –4, and so on.
___The one-to-one correspondence is given by n –(2n – 1).
___The one-to-one correspondence is given by n –(n – 1).
c) The integers with absolute value less than 1,000,000
___The set is countably infinite.
___The set is finite.
___The set is countably infinite with one-to-one correspondence 1 1,999,999, 2 1,999,998, 3 1,999,997, and so on.
___The set is countably infinite with one-to-one correspondence 0 1,999,999, 1 1,999,998, 2 1,999,997, and so on.
___The cardinality of the set is 1,999,999.
d) The real numbers between 0 and 2
___The set is countably infinite with one-to-one correspondence 1 0, 2 0.00001,3 0.00002, and so on.
___The set is uncountable.
___The set is finite.
___The set is countably infinite with one-to-one correspondence 1 0.00001, 20.00002, 30.00003, and so on.
e) The set A × Z+ where A = {2, 3}
___The set is countable.
___The set is countably infinite with one-to-one correspondence 1 (2,1), 2 (3,1),3 (2,2), 4 (3,2),and so on.
___The set is countably infinite with one-to-one correspondence 0 (2,1),1 (3,1),2 (2,2), 3 (3,2), and so on.
___The set is uncountable.
Explanation / Answer
(a) The set is countably infinite.
The set is countably infinite with one-to-one correspondence 1 11, 2 12, 3 13, and so on.
The one-to-one correspondence is given by n (n + 10).
The other correspondence between Z+ and this set is also one-one
(b)The set is countably infinite. The set is countably infinite with one-to-one correspondence 1 –1, 2 –3, 3 –5, 4 –7, and so on. The one-to-one correspondence is given by n –(2n – 1).
(c) The set is finite.
(d)The set is uncountable.
(e)The set is countably infinite with one-to-one correspondence 1 (2,1), 2 (3,1),3 (2,2), 4 (3,2),and so on.
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