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infinite set

burak100
in Analysis and Calculus

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f:R-->Q

 

show that there is a q \in Q such that f^-1(q) is infinite set in R

 

?????

f:R-->Q

 

show that there is a q \in Q such that f^-1(q) is infinite set in R

 

?????

 

 

This appears to be homework.

 

What do you know about cardinality ? This is a relatively simple exercise in the theory of cardinal numbers.

  • Author

Ok. I think; we have 

\mathbb{R}= \bigcup\limits_{q \in \mathbb{Q}} f^{-1}(q)

, and we suppose that for all q in Q , 

f^{-1}(q)

finite, then union would be countable but R is uncountable so contradiction... right?ITEX

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