Which Set Contains More Real Numbers: (-1, 1) or (1, 1,000,000)?

Comparing the Cardinality of Real Numbers in Different Intervals

The sets (−1,1) and (1,1,000,000)contain the same number of real numbers, as both are continuous intervals and have the same cardinality, which is the cardinality of the continuum (c).

Detailed Explanation

1. The Set (−1,1)
   • This open interval includes all real numbers strictly between −1 and 1.
   • It is a continuous set of real numbers, making it uncountably infinite, with a cardinality of (c).
2. The Set (1,1,000,000)
   • This open interval includes all real numbers strictly between 1 and 1,000,000.
   • Like the previous set, it is also a continuous set, uncountably infinite, and has a cardinality of (c).

Proof via Mapping

We can construct a bijection (one-to-one and onto function) between these two intervals. For example, consider the:

f(x)=999999​/2⋅(x+1)+1

• When x=−1, f(x)=1
• When x=1, f(x)=1,000,000

This function maps every real number in (−1,1) uniquely to a real number in (1,1,000,000), and vice versa. Since a bijection exists, the two sets have the same cardinality.

While the interval (1, 1,000,000) is “longer” in terms of its length on the real number line, the number of real numbers within it is the same as the number of real numbers in the much smaller interval (-1, 1). They are both uncountably infinite.

Intervals and Infinity: Any open interval (a, b) where a < b contains an uncountably infinite number of real numbers. This is a fundamental concept in set theory and real analysis.