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Lets say the set A consists of all the polynomials. And C denote the cardinal of real numbers. 1.) is the card (A) less than or equal to c? 2.) is the card of range of any nonconstant function from A = C? 3.) the range of any nonconstant function which can be expressed as a difference of two functions from A has card = C?

Jus
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1 Answers1

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Hint:

  1. A countably infinite union of sets of cardinality $C$ also has cardinality $C$
  2. The set $\mathbb R^n$ has cardinality $C$, and so does any set with a bijecion to $\mathbb R^n$.
  3. The set $A$ can be written as a countably infinite union of sets which, using point 2., can be shown to be of cardinality $C$.
5xum
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  • Does it work the same also if the set of polynomials is changed to the set of differentiable functions or integrable functions? – Jus Apr 03 '17 at 01:48
  • @Jus It doesn't work the same way, no. The same argument cannot be made. But I suggest you ask a new question for that. – 5xum Apr 03 '17 at 11:00