Question

Which of the following sets are finite or infinite
(i) The set of months of a year
(ii) $\left\{ {1,2,3,....} \right\}$
(iii) $\left\{ {1,2,3,....,99,100} \right\}$
(iv) The set of positive integers greater than $100$
(v) The set of prime numbers less than $99$

Answer 1

Hint: For each part, start with representing the given information in the roaster form. Now check whether the number of elements in the set is countable or uncountable. If the elements of the set are uncountable then the set will be an infinite set. In the case of a countable number of elements in the set, the set is called finite. Use this information in each case separately to find the answer.

Complete step-by-step answer:
Here in this problem, we are given five sets of numbers and we need to determine whether these are finite sets or infinite sets.
Before solving this question, we must understand the concept of an infinite and finite set. Finite sets are the sets having a finite/countable number of members. Finite sets are also known as countable sets as they can be counted. The process will run out of elements to list if the elements of this set have a finite number of members. If a set is not finite, it is called an infinite set because the number of elements in that set is not countable and also we cannot represent it in Roster form. Thus, infinite sets are also known as uncountable sets.
So, the elements of an Infinite set are represented by 3 dots (ellipse) thus, it represents the infinity of that set.
For part (i), the set of months of a year will contain all the twelve months that we have in a year. Thus the set can be represented as:
$ \Rightarrow A = \left\{ {Jan,Feb,Mar,Apr,May,Jun,Jul,Aug,Sep,Oct,Nov,Dec} \right\}$
So, it forms a set of countable elements with a number of elements of $12$ .
Hence, set (i) is a finite set.
For part (ii), the set $\left\{ {1,2,3,....} \right\}$ represents all the natural numbers and does not consist of any ending element.
This makes this set an uncountable set.
Hence, set (ii) is an infinite set.
For part (iii), the set $\left\{ {1,2,3,....,99,100} \right\}$ represents all the natural numbers from $1{\text{ to }}100$ .
The number of elements in the above set is $100$ and hence make this set a countable set.
Hence, set (iii) is a finite set.
For part (iv), the set of positive integers greater than $100$ will be a set with starting number as $101$ and with no ending element.
The set of positive integers greater than $100$ will be written as $\left\{ {101,102,103,.....} \right\}$ , which is an uncountable set.
Hence, set (iv) is an infinite set.
For part (v), the set of prime numbers less than $99$ is the set of natural numbers which are prime and also have an ending element less than $99$.
This set is represented as $\left\{ {2,3,5,7,11......,89,97} \right\}$ , which consists of a countable number of elements.
Hence, set (v) is a finite set.

Note: The contents of a set can be described by listing the elements of the set, separated by commas, inside a set of curly brackets. This way of describing a set is called the roster form. Using the roster form to represent a set was a crucial part of the solution. Remember when the set of a sequence is having both starting and ending elements, then it is a finite set.