Hi people, I need some help with these questions please:(adsbygoogle = window.adsbygoogle || []).push({});

1.Is the set of all x in the real numbers such that (x+pi) is

rational, countable?

I don't think this is countable, isn't the only possible value for x = -pi, all other irrationals will not make x+pi rational i thought?

2.Is the set of all x in the real numbers, such that for all k, (x+the square root of k) is not a natural number, countable?

Again I don't think this is countable because if the square root of k is irrational, like it is for k=2, then you can add infinitely many values of x to root k for which the sum of x and root k is not natural.

Lastly, is every infinite subset of the power set of the naturals uncountable?

I don't think so because P(N), the power set of N, has cardinality 2^(aleph nought) and is countable, hence it's subsets will be countable.

Could anyone offer some advice,

Cheers,

JB

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# Countable sets

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