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Q1(iv):
Which of the following are sets ? Justify your answer. (iv) The collection of all boys in your class.
Solution :
Step 1: Definition of a Set
According to Cantor’s set theory, a set is defined as a well-defined collection of objects [Definition per Set Theory Axioms]. The term "well-defined" implies that for any given object $x$ and a collection $A$, there exists an unambiguous rule to determine whether $x \in A$ or $x \notin A$.
Step 2: Analysis of the Provided Collection
The collection in question is: "The collection of all boys in your class."
- Let $C$ represent the specific classroom of interest.
- Let $U$ represent the universal set of all humans.
- The collection $B$ is defined by the predicate $P(x)$, where $P(x)$ is true if $x$ is a student in class $C$ AND $x$ is of the male gender.
Step 3: Verification of Well-Definedness
To determine if the collection is a set, we test the criterion of objectivity:
| Criterion | Application |
|---|---|
| Objectivity | The roster of a specific class is fixed and verifiable by school administration records. |
| Membership | For any individual person, one can definitively categorize them as either being in the class or not, and as being a boy or not. |
Since the criteria for membership are based on factual attributes (enrollment status and gender identification as recorded in institutional documentation), the collection is not subject to individual opinion, ambiguity, or variable interpretation [Ref: Axiom of Specification].
Visual Representation of Set Membership
Step 4: Conclusion
Because the collection is well-defined and the boundary for membership is clear, fixed, and objective, it satisfies the fundamental criteria of a mathematical set.
Final Solution: The collection of all boys in your class is a set, because it is a well-defined collection of objects where membership is objective and verifiable.
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(iii): Which of the following are sets ? Justify your answer. (iii) A team of eleven best-cricket batsmen of the world.
- 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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