Question

Consider these intervals as sets or real numbers (0, 1), (-1, 0), (0,5). Which statement below is true about their cardinality?

A. these are finite sets
B. all have the same cardinality aleph 1
C. all are counatable
D. we cannot find 1:1 correspondences between these sets
E. cardinality of (-1, 0) is the smallest.
F. all have the same cardinality aleph
G. cardinality of (0, 1) is the smallest.
H. cardinality of (0, 1) is less than cardinality of (0,5).

Answer 1

All intervals (0, 1), (-1, 0), and (0, 5) have the same uncountable cardinality, which is aleph one (ℕ1), not aleph null. Therefore, the correct answer is that all intervals have the same cardinality and that cardinality is not countable.

Step-by-step explanation:

The correct statement regarding the cardinality of the intervals as sets of real numbers (0, 1), (-1, 0), and (0, 5) is B. all have the same cardinality aleph null (also denoted as ℕ0). This is because each of these sets is an interval of real numbers, and between any two distinct real numbers, there are infinitely many other real numbers. Furthermore, each of these intervals can be put into a one-to-one correspondence with the set of all real numbers, meaning that they all have the cardinality of the continuum, ℕ1, which is greater than aleph null and is not countable. Therefore, options A, C, D, E, and G are incorrect, and since all intervals can be paired with each other through one-to-one correspondences, option H is also incorrect.