Exercise 1.3 — Sets

1. 1.3.12 marksNCERT Class 11 Mathematics, Sets, Reprint 2026-27

   Fill in the blank with the correct symbol (subset or not-a-subset): (i) {2,3,4}...{1,2,3,4,5} (ii) {a,b,c}...{b,c,d} (iii) {students of Class XI of your school}...{students of your school} (iv) {circles in the plane}...{circles in the plane with radius 1} (v) {triangles in a plane}...{rectangles in the plane} (vi) {equilateral triangles}...{triangles} (vii) {even naturals}...{integers}.

   Hint. Check whether EVERY element of the left-hand set is guaranteed to be in the right-hand set.

   (i) every listed number of the first set appears in the second. (ii) a is not in the second set. (iii) every Class XI student is a student of the school. (iv) most circles do not have radius exactly 1. (v) triangles are not rectangles. (vi) every equilateral triangle is, in particular, a triangle. (vii) every even natural number is an integer.

   ✦ Working through each part gives: (i) subset (ii) not a subset (iii) subset (iv) not a subset (v) not a subset (vi) subset (vii) subset.

2. 1.3.22 marksNCERT Class 11 Mathematics, Sets, Reprint 2026-27

   Examine whether each statement is true or false: (i) {a,b} is not a subset of {b,c,a}, (ii) {a,e} is a subset of {vowels}, (iii) {1,2,3} is a subset of {1,3,5}, (iv) {a} is a subset of {a,b,c}, (v) {a} is an element of {a,b,c}, (vi) {even naturals less than 6} is a subset of {naturals dividing 36}.

   Hint. Distinguish carefully between an object being an ELEMENT of a set and being a SUBSET of it -- (v) tests exactly this.

   (i) both a and b are in {b,c,a}, so {a,b} IS a subset, making the claim false. (ii) a,e are both vowels. (iii) 2 is not in {1,3,5}. (iv) a is in {a,b,c}. (v) {a,b,c} has elements a, b, c -- not the set {a} -- so {a} is not literally an element (only 'a' is). (vi) {2,4} both divide 36.

   ✦ Working through each part gives: (i) False (ii) True (iii) False (iv) True (v) False (vi) True.

   Where students slip. Confusing 'a is an element of the set' with '{a} is an element of the set' is the exact trap in part (v).

3. 1.3.33 marksNCERT Class 11 Mathematics, Sets, Reprint 2026-27

   Let A = {1, 2, {3,4}, 5}. Which of the following are incorrect, and why? (i) {3,4} subset A (ii) {3,4} in A (iii) {{3,4}} subset A (iv) 1 in A (v) 1 subset A (vi) {1,2,5} subset A (vii) {1,2,5} in A (viii) {1,2,3} subset A (ix) phi in A (x) phi subset A (xi) {phi} subset A.

   Hint. A's actual elements are exactly four objects: 1, 2, the set {3,4} (as a single object), and 5. The number 3 alone is not an element of A.

   (i) incorrect -- 3 and 4 are not individually elements of A, so {3,4} cannot be a subset; it IS an element, not a subset. (ii) correct -- {3,4} is literally one of A's four elements. (iii) correct -- since {3,4} is an element of A, the set containing it is a subset. (iv) correct. (v) incorrect -- 1 is a number, not a set, so 'subset' does not apply; it should be 1 in A. (vi) correct -- 1,2,5 are each elements of A. (vii) incorrect -- {1,2,5} itself is not one of A's four listed elements. (viii) incorrect -- 3 alone is not an element of A. (ix) incorrect -- the empty set is not listed among A's elements. (x) correct -- phi is a subset of every set. (xi) incorrect -- {phi} would need phi itself to be an element of A, which it is not.

   ✦ Working through each part gives: incorrect: (i),(v),(vii),(viii),(ix),(xi). Correct: (ii),(iii),(iv),(vi),(x).

4. 1.3.42 marksNCERT Class 11 Mathematics, Sets, Reprint 2026-27

   Write down all the subsets of: (i) {a}, (ii) {a,b}, (iii) {1,2,3}, (iv) phi.

   Hint. A set with n elements has exactly 2^n subsets, including phi and the set itself -- list them systematically by how many elements each contains.

   (i) 2^1=2 subsets. (ii) 2^2=4 subsets. (iii) 2^3=8 subsets, taken zero, one, two, or three elements at a time. (iv) phi has only itself as a subset.

   ✦ Working through each part gives: (i) {phi,{a}} (ii) {phi,{a},{b},{a,b}} (iii) {phi,{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}} (iv) {phi}.

5. 1.3.52 marksNCERT Class 11 Mathematics, Sets, Reprint 2026-27

   Write the following as intervals: (i) {x in R : -4<x<=6} (ii) {x in R : -12<x<-10} (iii) {x in R : 0<=x<7} (iv) {x in R : 3<=x<=4}.

   Hint. A round bracket excludes the endpoint; a square bracket includes it.

   Read off directly which endpoints are included based on the inequality signs used.

   ✦ Working through each part gives: (i) (-4,6] (ii) (-12,-10) (iii) [0,7) (iv) [3,4].

6. 1.3.62 marksNCERT Class 11 Mathematics, Sets, Reprint 2026-27

   Write the following intervals in set-builder form: (i) (-3,0) (ii) [6,12] (iii) (6,12] (iv) [-23,5).

   Hint. Round brackets become strict inequalities; square brackets become 'less than or equal to' / 'greater than or equal to'.

   Translate each bracket directly into the matching inequality symbol.

   ✦ Working through each part gives: (i) {x in R:-3<x<0} (ii) {x in R:6<=x<=12} (iii) {x in R:6<x<=12} (iv) {x in R:-23<=x<5}.

7. 1.3.72 marksNCERT Class 11 Mathematics, Sets, Reprint 2026-27

   What universal set would you propose for: (i) the set of right triangles, (ii) the set of isosceles triangles?

   Hint. A universal set only needs to be broad enough to contain every object under discussion in that context.

   Both right triangles and isosceles triangles are, in particular, triangles, so the natural, smallest sensible universal set for either is the set of all triangles in a plane.

   ✦ Working through each part gives: (i) and (ii): the set of all triangles in a plane.

8. 1.3.82 marksNCERT Class 11 Mathematics, Sets, Reprint 2026-27

   Given A={1,3,5}, B={2,4,6}, C={0,2,4,6,8}, which of {0,1,2,3,4,5,6}, phi, {0,1,...,10}, {1,2,...,8} may be a universal set for A, B and C together?

   Hint. A valid universal set must contain every element of A, B, and C simultaneously.

   {0,...,6} is missing 8 (needed for C). phi contains nothing. {0,...,10} contains every element of A, B and C. {1,...,8} is missing 0 (needed for C).

   ✦ Working through each part gives: only {0,1,2,...,9,10} works as a universal set for A, B and C.

Solutions written by the tuition.in editorial team and checked against the NCERT Class 11 Mathematics textbook, Reprint 2026-27 (kemh101.pdf) — five numbered exercises (1.1-1.5, 39 questions total) plus the chapter's Miscellaneous Exercise (10 questions); every question solved directly from the given sets rather than from memory, including the multi-part true/false and subset-vs-membership questions in Exercise 1.3 Q3 and the Miscellaneous Exercise Q2, which require care to avoid conflating 'element of' with 'subset of'. Questions are referenced from the NCERT textbook for identification.