INTRODUCTION
Everything present in this universe living or non-living is called an object.
A set is a well-defined collection of objects. In other words a set is a collection of objects which are well-defined. A set fullfil both conditions: 1. It is a collection of objects & 2. Objects should be well-defined.
Well-defined means, it should be very
clear that the object belongs to a collection or not.
We denote the set with
capital letters like A, B, C, P, Q, R, X, Y, Z, etc.
If a is an element of a set A, then
we can say that a belongs to A and represent a ∈ A.
If a is not an element of a
set A, then we can say that a doesn't belong to A and represent a ∉ A.
The objects in a set are called elements, points, or members. If a ∈ A, then a is an element of set A.
All the members of the set are written within braces { } and separated by commas.
ILLUSTRATIONS
- Collection of famous poets. Here objects/elements are poets, but famous is not well defined as famous changes as per observer.
- The collection of all vowels in the English alphabet. It contains five elements, namely a, e, i, o, u and this collection is well-defined and therefore, it is a set.
- The collection of all even natural numbers less than 15. It contains the numbers 2, 4, 6, 8, 10, 12, 14 and the collection is well-defined therefore, it is a set.
- All possible roots of the quadratic equation x2 + 2x - 3= 0 are 1 and - 3. The collection of all possible roots of x2 + 2x - 3= 0 (i.e. 1 and -3) is well defined and therefore, it is a set.
- The collection of all cities of India. It is clearly well-defined and therefore, it is a set.
- The collection of the five most talented writers of India is not a set since no rule has been given for deciding whether a given writer is talented or not.
EXERCISE 1A
Question 1. Which of the following are sets? Justify your answer.
Question 1(i) The collection of all whole numbers less than 10.
Solution: Collection of all whole numbers less than 10 are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. It is well defined as less than 10. So, it is a set.
Question 1(ii) The collection of good hockey players in India.
Solution: Collection of hockey players, but who are good. It is not well defined as good players are different for different observers. So, the given collection is not a set.
Question 1(iii) The collection of all questions in this chapter.
Solution: The collection of all questions in this chapter is a well-defined collection. All questions belong to this chapter. So, it is a set.
Question 1(iv) The collection of all difficult chapters in this book.
Solution: The collection of difficult chapters, but this is not defined on which basis difficult criteria based, it is not well-defined. Therefore, it is not a set.
Question 1(v) A collection of Hindi novels written by Munshi Prem Chand.
Solution: Here collection of objects is a collection of Hindi novels. And are well-defined as written by Munshi Prem Chand. So, the given collection is a set.
Question 1(vi) A team of 11 best cricket players of India.
Solution: Best players changed from time to time and no criteria is given to select the best player, so it is not a well-defined collection. Therefore the collection is not a set.
Question 1(vii) The collection of all the months of the year whose names begin with the letter M.
Solution: It is a well-defined collection of all the months of the year whose names begin with the letter M, namely March and May. So, this collection is a set.
Question 1(viii) The collection of all interesting books.
Solution: It is not defined how interesting books are judged. So, this collection is not a set.
Question 1(ix) The collection of all short boys of your class.
Solution: It is not defined how short boys are selected. So, the given collection is not a set.
Question 1(x) The collection of all those students of your class whose ages exceed 15 years.
Solution: It is a well-defined collection of all those students in your class whose ages exceed 15 years. So, it is a set.
Question 1(xi) The collection of all rich persons of Kolkata.
Solution: On which criteria are rich persons selected are not given. So, it is not a set.
Question 1(xii) The collection of all persons of Kolkata whose assessed annual incomes exceed (say) 20 lakh in the financial year 2016–17.
Solution: The given collection is clearly well-defined. So, it is a set.
Question 1(xiii) The collection of all interesting dramas written by Shakespeare.
Solution: The term ‘interesting‘ is not defined. So, it is not a set.