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No. There are infinitely many rational numbers between any two integers.

Q: Is it possible to count the number of rational numbers there are between any two integers?

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The set of integers is a proper subset of the set of rational numbers.

All integers are rational numbers. There are integers with an i behind them that are imaginary numbers. They are not real numbers but they are rational. The square root of 2 is irrational. It is real but irrational.

Yes, but there are countably infinite such numbers.

Fractions are not integers. They may or may not be rational numbers.

Counting numbers are a proper subset of whole numbers which are the same as integers which are a proper subset of rational numbers.

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The set of integers is a proper subset of the set of rational numbers.

All integers are rational numbers. There are integers with an i behind them that are imaginary numbers. They are not real numbers but they are rational. The square root of 2 is irrational. It is real but irrational.

A.(Integers) (Rational numbers)B.(Rational numbers) (Integers)C.(Integers) (Rational numbers)D.(Rational numbers) (Real numbers)

Yes, but there are countably infinite such numbers.

Integers are aproper subset of rational numbers.

Fractions are not integers. They may or may not be rational numbers.

Counting numbers are a proper subset of whole numbers which are the same as integers which are a proper subset of rational numbers.

All integers are rational numbers.

Rational numbers are integers and fractions

No, integers are a subset of rational numbers.

If I understand your question, the answer is 'no', because all integers are rational numbers.

They can be integers, rational numbers or even approximations for irrational numbers.