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Q1(iii):
Which of the following are sets ? Justify your answer. (iii) A team of eleven best-cricket batsmen of the world.
Solution :
Definition and Theoretical Foundation
A set is defined as a well-defined collection of objects. The condition of "well-defined" implies that for any given object, it must be possible to state definitively whether it belongs to the collection or not, based on a specific, objective criterion. If the criteria for inclusion are subjective, ambiguous, or dependent on individual opinion, the collection does not constitute a set [Per Cantor’s Set Theory].
Analysis of the Collection: "A team of eleven best-cricket batsmen of the world"
To determine if this collection is a set, we evaluate the term "best":
- The definition of "best" is subjective.
- One individual might define "best" based on the highest career batting average ($ \text{avg} $).
- Another individual might define "best" based on the highest strike rate ($ \text{SR} $) or total centuries ($ C $).
- There is no universally accepted, objective mathematical standard that identifies the same "eleven batsmen" for every observer.
Logical Justification
Since the membership of the collection depends entirely on the criteria selected by the observer, the collection lacks universality. Because the property of being "best" is not defined by an objective, invariant attribute, the collection is considered ill-defined. [As per the axioms of Set Theory, a set must be independent of the observer's opinion].
Comparative Analysis Table
| Criteria | Classification | Result |
|---|---|---|
| All cricket batsmen with $> 10,000$ runs in Test cricket | Well-defined | Set |
| A team of eleven best-cricket batsmen | Ill-defined (Subjective) | Not a Set |
Final Solution: The collection of "eleven best-cricket batsmen of the world" is not a set because the term "best" is subjective and does not provide an objective criterion for membership.
More Questions from Class 11 Mathematics Sets EXERCISE 1.1
- Q1(i): Which of the following are sets ? Justify your answer. (i) The collection of all the months of a year beginning with the letter J.
- Q1(ii): Which of the following are sets ? Justify your answer. (ii) The collection of ten most talented writers of India.
- Q1(iv): Which of the following are sets ? Justify your answer. (iv) The collection of all boys in your class.
- Q1(ix): Which of the following are sets ? Justify your answer. (ix) A collection of most dangerous animals of the world.
- Q1(v): Which of the following are sets ? Justify your answer. (v) The collection of all natural numbers less than 100.
- Q1(vi): Which of the following are sets ? Justify your answer. (vi) A collection of novels written by the writer Munshi Prem Chand.
- Q1(vii): Which of the following are sets ? Justify your answer. (vii) The collection of all even integers.
- Q1(viii): Which of the following are sets ? Justify your answer. (viii) The collection of questions in this Chapter.
- Q2(i): Let $A = \{1, 2, 3, 4, 5, 6\}$. Insert the appropriate symbol $\in$ or $\notin$ in the blank spaces: (i) 5. . .A
- Q2(ii): Let $A = \{1, 2, 3, 4, 5, 6\}$. Insert the appropriate symbol $\in$ or $\notin$ in the blank spaces: (ii) 8 . . . A
- Q2(iii): Let $A = \{1, 2, 3, 4, 5, 6\}$. Insert the appropriate symbol $\in$ or $\notin$ in the blank spaces: (iii) 0. . .A
- Q2(iv): Let $A = \{1, 2, 3, 4, 5, 6\}$. Insert the appropriate symbol $\in$ or $\notin$ in the blank spaces: (iv) 4. . . A
- Q2(v): Let $A = \{1, 2, 3, 4, 5, 6\}$. Insert the appropriate symbol $\in$ or $\notin$ in the blank spaces: (v) 2. . .A
- Q2(vi): Let $A = \{1, 2, 3, 4, 5, 6\}$. Insert the appropriate symbol $\in$ or $\notin$ in the blank spaces: (vi) 10. . .A
- Q3(i): Write the following sets in roster form: (i) $A = \{x : x$ is an integer and $–3 \le x < 7\}$
- Q3(ii): Write the following sets in roster form: (ii) $B = \{x : x$ is a natural number less than 6\}
- Q3(iii): Write the following sets in roster form: (iii) $C = \{x : x$ is a two-digit natural number such that the sum of its digits is 8\}
- Q3(iv): Write the following sets in roster form: (iv) $D = \{x : x$ is a prime number which is divisor of 60\}
- Q3(v): Write the following sets in roster form: (v) $E$ = The set of all letters in the word TRIGONOMETRY
- Q3(vi): Write the following sets in roster form: (vi) $F$ = The set of all letters in the word BETTER
- Q4(i): Write the following sets in the set-builder form : (i) $\{3, 6, 9, 12\}$
- Q4(ii): Write the following sets in the set-builder form : (ii) $\{2,4,8,16,32\}$
- Q4(iii): Write the following sets in the set-builder form : (iii) $\{5, 25, 125, 625\}$
- Q4(iv): Write the following sets in the set-builder form : (iv) $\{2, 4, 6, . . .\}$
- Q4(v): Write the following sets in the set-builder form : (v) $\{1,4,9, . . .,100\}$
- Q5(i): List all the elements of the following sets : (i) $A = \{x : x$ is an odd natural number\}
- Q5(ii): List all the elements of the following sets : (ii) $B = \{x : x$ is an integer, $– \frac{1}{2} < x < \frac{9}{2}\}$
- Q5(iii): List all the elements of the following sets : (iii) $C = \{x : x$ is an integer, $x^2 \le 4\}$
- Q5(iv): List all the elements of the following sets : (iv) $D = \{x : x$ is a letter in the word “LOYAL”\}
- Q5(v): List all the elements of the following sets : (v) $E = \{x : x$ is a month of a year not having 31 days\}
- Q5(vi): List all the elements of the following sets : (vi) $F = \{x : x$ is a consonant in the English alphabet which precedes $k$ \}.
- Q6: Match each of the set on the left in the roster form with the same set on the right described in set-builder form: (i) $\{1, 2, 3, 6\}$ (a) $\{x : x$ is a prime number and a divisor of 6\} (ii) $\{2, 3\}$ (b) $\{x : x$ is an odd natural number less than 10\} (iii) $\{M,A,T,H,E,I,C,S\}$ (c) $\{x : x$ is natural number and divisor of 6\} (iv) $\{1, 3, 5, 7, 9\}$ (d) $\{x : x$ is a letter of the word MATHEMATICS\}.
CBSE Solutions for Class 11 Mathematics Sets
Chapters in CBSE - Class 11 Mathematics
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