Part BCSIR NET June 2024interval-has-cardinality-of-R

Interval has cardinality of R

Let S = { $x \in R : x > 1$ and $(1 - x^{4}) / (1 - x^{3}) > 22$ }. Which of the following is true about S?

A.S is empty.
B.There is a bijection between S and $N$
C.There is a bijection between S and $R$ ✓
D.There is a bijection between S and a non-empty finite set

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward.

50 are analysed free — try those first.

Related counterexample: Every ordered field is Archimedean

More on this topic

on a finite set every function is continuousDecember 2024
only an interval condition leaves uncountably manyDecember 2024
a connected domain forces a constant mapDecember 2024
the supremum is approached and not attainedDecember 2024
sup need not be attainedJune 2024
cantor no set of that sizeJune 2024

The chapter behind this: The real line: supremum, Archimedes and density — free to read

From The Real Line › Completeness, sup/inf, Archimedean property

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