Chapter R, Problem 1T

To determine

To state: Whether the set of negative integers is finite or infinite by listing the elements of the set.

Answer to Problem 1T

The set of negative integers is $\underset{\_}{\text{infinite}}$ as in listing form set is $\underset{\_}{\left\{-1,-2,-3,-4,-5,\mathrm{...}\right\}}$.

Explanation of Solution

Given:

The set of negative integer’s element.

Definition used:

Integers are the set of whole numbers or numbers that are not fractions.

The natural numbers is the set of numbers which is defined as $N=\left\{1,2,3,4,\mathrm{...}\right\}$, where three dots indicate that the set continues indefinitely in a sequence.

Two type of sets are possible on the basis of counting, one that has finite number of elements and one that has infinite number of elements.

Also in infinite set, there are two possible types one that can be listed in elements form and one that can’t be listed in elements form.

Calculation:

Negative integers are the numbers that are obtained by negating the natural numbers. The set of negative integers is as follows:

$Z^{-}=\left\{-1,-2,-3,-4,-5,\mathrm{...}\right\}$

Here, $Z^{-}$ denote the set of negative integers.

By the definition stated above, set of natural number $N$ is an infinite set.

Therefore, the set of negative integers will also be an infinite set as it is obtained by negating the natural numbers.

Thus, the set of negative integers is $\underset{\_}{\text{infinite}}$ as in listing form set is $\underset{\_}{\left\{-1,-2,-3,-4,-5,\mathrm{...}\right\}}$.