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Theory of Positive Integers : Theory of Positive Integers

Study concepts, example questions & explanations for Theory of Positive Integers

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Example Questions

Example Question #1 : Cardinality

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

Possible Answers:

Countably

Negatively

None of the answers

Continuously

Positively

Correct answer:

Countably

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 "countably".

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Example Question #1 : Cardinality

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

Possible Answers:

Negatively

Uncountably

None of the answers

Positively

Infinitely

Correct answer:

Uncountably

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 "uncountably".

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Theory of Positive Integers : Theory of Positive Integers

Study concepts, example questions & explanations for Theory of Positive Integers

All Theory of Positive Integers Resources

Example Questions

Example Question #1 : Cardinality

Example Question #1 : Cardinality

All Theory of Positive Integers Resources

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