Write down descriptions for each of the following sets:

● Three different countably infinite subsets of the real numbers.

● Three different uncountably infinite subsets of the real numbers.

● One uncountably infinite proper super set of the real numbers.

Solutions

Expert Solution

Three countably infinite subsets of the real numbers are:

The set of all natural numbers.
The set of all even integers.
The set of all rational numbers.

Three uncountably infinite subsets of the real numbers are:

The set of all irrational numbers.
The set of all real numbers in the open interval (0,1).
The set of all real numbers in the closed interval [0,1] .

An uncountably infinite proper super set of the real numbers is:

The union of the set of real numbers with the set {Chegg}.
Explanation: This set contains all the elements of the set of real numbers, hence it is a superset of the real numbers. It contains an element Chegg which is not in the set of real numbers hence it is a proper subset. Also, it is an infinite set, because it has a subset (the set of real numbers) which is infinite. Finally, it is uncountable, because the set of real numbers have no bijection with the set of natural numbers.

venereology answered 1 year ago

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