Question

prove or disprove

1. A Turing machine with two tapes is no more powerful than a Turing machine with one tape. (That is, both types of machines can compute the same set of functions.)
2. The cardinality of the set of irrational numbers is greater than the cardinality of the set of all rational numbers.
3. The cardinality of the set of all algebraic numbers is exactly the same as the cardinality of all real numbers.

Answer 1

First statement is True. "A Turing machine with two tapes is no more powerful than a Turing machine with one tape." Though having two tapes it seems more powerful than single tape turning machines, but still they can be simulated by single taped turning machines and cannot calculate any more functions than single taped machines.

Second statement is also True. "The cardinality of the set of irrational numbers is greater than the cardinality of the set of all rational numbers." As cardinality of rational numbers is equal to that of natural numbers and hence is countable. If cardinality of irrational will also be countable then union of both rational and irrational will be countable too which is not. So, cardinality of irrational numbers is not countable. Hence, the cardinality of the set of irrational numbers is greater than the cardinality of the set of all rational numbers.

Third statement is False. We know that cardinality of transcendental numbers is same as that of the real numbers (Transcendental numbers are those which aren't algebraic numbers). And the point is that there are less number of algebraic numbers than the transcendental numbers. Hence the statement "The cardinality of the set of all algebraic numbers is exactly the same as the cardinality of all real numbers" is False.

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