Where Mindful Learning Begins
Where Mindful Learning Begins
Free, step-by-step Class 11 Maths NCERT Solutions for Chapter 1 Ex 1.1 — all 6 questions solved, covering well-defined collections, the ∈ and ∉ symbols, and converting sets between roster form and set-builder form.
Question 1 sets up the whole chapter's core idea — a collection is only a set if it's well-defined, so subjective phrases like "most talented" or "most dangerous" get ruled out while objective ones don't. Question 2 is a quick warm-up on membership notation, and Questions 3–5 are where most of the practice lies: moving fluently between roster form (listing elements) and set-builder form (describing a property), including two-digit number puzzles, word-letter sets, and inequality-based integer sets. Question 6 closes the exercise by asking you to match roster-form sets to their set-builder description, which is really just Questions 3–4 in reverse.
A collection is a set only if it is well-defined, i.e., we can say for certain whether a given object belongs to it or not.
(i) Months beginning with J: January, June, July — a fixed, unambiguous list.
∴ It is a set.
(ii) "Most talented" is a matter of opinion; it varies from person to person.
∴ It is not a set.
(iii) "Best" cricket batsmen is subjective and differs from one judge to another.
∴ It is not a set.
(iv) Every boy in the class is a definite, identifiable individual.
∴ It is a set.
(v) Natural numbers less than 100 are precisely 1, 2, 3, ..., 99.
∴ It is a set.
(vi) The novels written by Munshi Prem Chand form a fixed, identifiable collection.
∴ It is a set.
(vii) Even integers are precisely defined: ..., −4, −2, 0, 2, 4, ...
∴ It is a set.
(viii) The questions in this chapter form a fixed, countable collection.
∴ It is a set.
(ix) "Most dangerous" is subjective — it varies from person to person.
∴ It is not a set.
Given A=\{1,2,3,4,5,6\}:
(i) 5 is an element of A.
5\in A
(ii) 8 is not an element of A.
8\notin A
(iii) 0 is not an element of A.
0\notin A
(iv) 4 is an element of A.
4\in A
(v) 2 is an element of A.
2\in A
(vi) 10 is not an element of A.
10\notin A
Listing every integer from −3 up to (but not including) 7:
Checking two-digit numbers whose digits add to 8: 17, 26, 35, 44, 53, 62, 71, 80.
Since 60=2^2\times3\times5, its only prime divisors are 2, 3 and 5.
Listing each distinct letter once (repeated letters like R, O, T are not repeated in a set):
The distinct letters appearing are B, E, T and R.
(i) Each term is 3 times a natural number, from n=1 to n=4.
(ii) Each term is a power of 2, from 2^1 to 2^5.
(iii) Each term is a power of 5, from 5^1 to 5^4.
(iv) The pattern continues indefinitely — this is the set of all even natural numbers.
(v) Each term is the square of a natural number, from 1^2 to 10^2.
(i) The odd natural numbers continue indefinitely, so A is infinite.
(ii) \tfrac{9}{2}=4.5, so the integers strictly between -0.5 and 4.5 are 0, 1, 2, 3, 4.
(iii) x^2\le4 \Rightarrow -2\le x\le2, so the integers satisfying this are −2, −1, 0, 1, 2.
(iv) The distinct letters in "LOYAL" are L, O, Y, A (L is not repeated in a set).
(v) The months without 31 days are April, June, September, November (30 days each), and February (28 or 29 days).
(vi) The consonants before k in the English alphabet are b, c, d, f, g, h, j.
(i) \{1,2,3,6\} are the natural-number divisors of 6.
∴ matches (c).
(ii) \{2,3\} are the divisors of 6 that are also prime.
∴ matches (a).
(iii) The distinct letters of "MATHEMATICS" are M, A, T, H, E, I, C, S.
∴ matches (d).
(iv) \{1,3,5,7,9\} are the odd natural numbers less than 10.
∴ matches (b).
| (i) {1, 2, 3, 6} | → | (c) divisors of 6 |
| (ii) {2, 3} | → | (a) prime divisors of 6 |
| (iii) {M,A,T,H,E,I,C,S} | → | (d) letters of MATHEMATICS |
| (iv) {1, 3, 5, 7, 9} | → | (b) odd naturals < 10 |
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