Too much denseness!
Which of the following statements are true:
I. Between any two rational numbers , there always exists an irrational number.
II. Between any two irrational numbers , there always exists a rational number.
III. The size of the rational number set is equal to the size of the irrational number set.
IV. The size of the rational number set is equal to the size of the integer number set.
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1 solution
A quick outline: The rational numbers are dense in the reals, so I and II hold. The cardinality of the rational numbers, which is also the cardinality of the integers by a diagonalization argument, is ℵ 0 , while the reals have cardinality 2 ℵ 0 , so additionally only IV holds. Please refer to the following resources: https://proofwiki.org/wiki/Rationals are Everywhere Dense in_Reals Card Q = Card N : https://natureofmathematics.wordpress.com/lecture-notes/cantor/