Discussion
Let \(N\) be the set of natural numbers. Consider the following sets.
\(P\): Set of Rational numbers (positive and negative)
\(Q\): Set of functions from {0, 1} to \(N\)
\(R\): Set of functions from \(N\) to {0, 1}
\(S\): Set of finite subsets of \(N\).
Which of the sets above are countable?
Answer: D
The countable sets are P, Q and S.
- P: the rationals are countable, since each is a pair of integers and the pairs can be listed.
- Q: a function from {0, 1} to N is just an ordered pair of natural numbers, and N × N is countable.
- S: the finite subsets of N can be listed by size and largest element, or matched with the natural numbers through binary representation.
- R: a function from N to {0, 1} is an infinite binary sequence, and Cantor's diagonal argument shows these are uncountable.
So any option containing R, or leaving out one of P, Q or S, is wrong.
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