Question

State which of the following collections are set:\[\]
(i) The collection of good students in your school:\[\]
(ii) The collection of the numbers between 30 and 45. :\[\]
(iii) The collection of interesting books in your school library. :\[\]
(iv) Last three days of a week. :\[\]

Answer 1

Hint: We recall the definition of the set and the idea of well-defined elements, which can only belong to a set. Whenever we see the adjectives to make a collection the collection cannot have elements. We use it to check whether a collection is set or not.

Complete step-by-step solution
We know that a set is a collection of well-defined objects that can be listed. The members of a set are called elements. The word well –defined means the elements either will belong to the set or they won’t belong at all. If there is a set $A$ then whether an element $a$ will belong in $A$ will be a mathematical statement with objective sense. If it is an element we write $a\in A$. If there are $n$ elements say ${{a}_{1}},{{a}_{2}},{{a}_{3}},...,{{a}_{n}}$ we can write $A$ in list form or roster notation as
\[A=\left\{ {{a}_{1}},{{a}_{2}},{{a}_{3}},...,{{a}_{n}} \right\}\]
(i) The collection of good students in your school :\[\]
The above collection will include good students in school. The word “good” is not well-defined as the definition of goodness maybe performance in academics, sports or just being disciplined which is subjective, not objective. So the collection cannot be set. :\[\]
(ii) The collection of the numbers between 30 and 45. :\[\]
The above collection will include all numbers between 30 and 45. The numbers either will belong to the collection or will not belong objectively. We can write the set in set-builder form as
\[A=\left\{ x:30 < x < 35 \right\}\]
(iii) The collection of interesting books in your school library. :\[\]
The above collection will include interesting books in the school library. The word ‘interesting ’ is an adjective, which is not well-defined because different persons will find different types of books interesting. So the collection is not set. :\[\]
(iv) Last three days of a week. :\[\]
We know there are 7 days a week: Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, and Saturday. The last three days of the week are Thursday, Friday, and Saturday. If we make a collection of the last three days, the rest days will not belong. So we have the set as
\[A=\left\{ \text{Thursday, Friday, Saturday} \right\}\]

Note: The collection with adjectives can be fuzzy sets, which are defined on the degree of belonging. The set we studied is a discrete set where an element will either belong (degree of belonging 1) or will not belong (degree of belonging 0) but in fuzzy sets, the degree of belonging of elements lies between 0 and 1.