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No, an integer is a whole number. For example, 2 is an integer but 2.25 is not. -48 is an integer but -48.5 is not.
Integer numbers are a subset of real numbers. Real numbers may contain fractions.
For example the set of all numbers which are integer multiples of 4 is a subset of all the numbers exactly divisible by 2.
Yes - in fact the set of all even numbers is a subset of the set of all integers, which is, in turn, a subset of the set of all real numbers.
Yes, just look at decimals. Note: integers are a subset of rational numbers.