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To which of the following statements can the transfer principle be applied? If you think it can't be applied to a certain statement, try to prove that the statement is false for the hyperreals, e.g., by giving a counterexample.
- For any real numbers and , .
- The sine of any real number is between -1 and 1.
- For any real number , there exists another real number that is greater than .
- For any real numbers , there exists another real number such that .
- For any real numbers , there exists a rational number such that . (A rational number is one that can be expressed as an integer divided by another integer.)
- For any real numbers , , and , .
- For any real numbers and , either or or
- For any real number , .
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