Theory of Positive Integers - Cardinality

Which of the following is a classification of a set whose cardinality is infinite?

Uncountably

Positively

Negatively

Infinitely

None of the answers

Explanation

If a set is stated to have infinite cardinality then it will fall one of the following categories,

I. Countably

II. Uncountably

Countably infinite sets are those that the elements within the set are able to be counted.

For example, the set of natural numbers

$\mathbb{N}:\begin{Bmatrix} 0,1,2,3,3,...,n \end{Bmatrix}$

is a countably infinite set.

Uncountably infinite sets are those that the elements cannot be counted.

For example, the set of real numbers,

$\mathbb{R}:\begin{Bmatrix} ...,-1,-\frac{2}{3},0.4567786,5,\frac{145}{2},... \end{Bmatrix}$

The reason the set of real numbers is not countable is because there are infinitely many elements between each element. This differs from the natural numbers because in the natural numbers there are no elements between elements.

Therefore the correct solution is "uncountably".

Which of the following is a classification of a set whose cardinality is infinite?

Countably

Positively

Negatively

Continuously

None of the answers

Explanation

If a set is stated to have infinite cardinality then it will fall one of the following categories,

I. Countably

II. Uncountably

Countably infinite sets are those that the elements within the set are able to be counted.

For example, the set of natural numbers

$\mathbb{N}:\begin{Bmatrix} 0,1,2,3,3,...,n \end{Bmatrix}$

is a countably infinite set.

Uncountably infinite sets are those that the elements cannot be counted.

For example, the set of real numbers,

$\mathbb{R}:\begin{Bmatrix} ...,-1,-\frac{2}{3},0.4567786,5,\frac{145}{2},... \end{Bmatrix}$

Therefore the correct solution is "countably".

Cardinality

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