Exercise 1.4 — Sets

1. 1.4.12 marksNCERT Class 11 Mathematics, Sets, Reprint 2026-27

   Find the union of each pair: (i) X={1,3,5},Y={1,2,3} (ii) A={a,e,i,o,u},B={a,b,c} (iii) A={naturals that are multiples of 3},B={naturals less than 6} (iv) A={naturals,1<x<=6},B={naturals,6<x<10} (v) A={1,2,3},B=phi.

   Hint. The union collects every element that appears in at least one of the two sets, listed once each.

   Combine each pair's elements, dropping duplicates: (i) merges to five numbers. (ii) merges to seven letters. (iii) combines {1,2,3,4,5} with all multiples of 3. (iv) the two ranges join into one continuous block, 2 through 9. (v) union with the empty set changes nothing.

   ✦ Working through each part gives: (i) {1,2,3,5} (ii) {a,b,c,e,i,o,u} (iii) {1,2,3,4,5} union {multiples of 3} (iv) {2,3,4,5,6,7,8,9} (v) {1,2,3}.

2. 1.4.21 markNCERT Class 11 Mathematics, Sets, Reprint 2026-27

   Let A={a,b}, B={a,b,c}. Is A subset of B? What is A union B?

   Hint. Check every element of A against B first, then form the union.

   Both a and b are in B, so A is a subset of B. When A is a subset of B, their union is simply B itself, since A contributes nothing new.

   ✦ Working through each part gives: a subset B is true; A union B = B = {a,b,c}.

3. 1.4.31 markNCERT Class 11 Mathematics, Sets, Reprint 2026-27

   If A and B are two sets such that A subset B, what is A union B?

   Hint. If every element of A already sits inside B, adding A's elements to B changes nothing.

   Since A contributes no element outside B, the union collapses to exactly B.

   ✦ Working through each part gives: a union B = B whenever A is a subset of B.

4. 1.4.43 marksNCERT Class 11 Mathematics, Sets, Reprint 2026-27

   If A={1,2,3,4}, B={3,4,5,6}, C={5,6,7,8}, D={7,8,9,10}, find (i) A union B (ii) A union C (iii) B union C (iv) B union D (v) A union B union C (vi) A union B union D (vii) B union C union D.

   Hint. Merge the listed sets pairwise or all at once, dropping repeats.

   Combine elements systematically from left to right for each requested union.

   ✦ Working through each part gives: (i) {1,2,3,4,5,6} (ii) {1,2,3,4,5,6,7,8} (iii) {3,4,5,6,7,8} (iv) {3,4,5,6,7,8,9,10} (v) {1,2,3,4,5,6,7,8} (vi) {1,2,3,4,5,6,7,8,9,10} (vii) {3,4,5,6,7,8,9,10}.

5. 1.4.52 marksNCERT Class 11 Mathematics, Sets, Reprint 2026-27

   Find the intersection of each pair of sets from question 1 above.

   Hint. The intersection keeps only elements common to both sets in each pair.

   Compare each pair from question 1 and retain only shared elements: (i) X and Y share two numbers. (ii) A and B share only the letter a. (iii) multiples of 3 among the naturals below 6. (iv) the two ranges 2-6 and 7-9 share nothing. (v) intersecting with the empty set always gives the empty set.

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

6. 1.4.63 marksNCERT Class 11 Mathematics, Sets, Reprint 2026-27

   If A={3,5,7,9,11}, B={7,9,11,13}, C={11,13,15}, D={15,17}, find (i) A cap B (ii) B cap C (iii) A cap C cap D (iv) A cap C (v) B cap D (vi) A cap (B cup C) (vii) A cap D (viii) A cap (B cup D) (ix) (A cap B) cap (B cup C) (x) (A cup D) cap (B cup C).

   Hint. Work out any needed unions first, then intersect -- follow the order the brackets dictate.

   (i) numbers in both A and B. (ii) numbers in both B and C. (iii) A cap C is {11}, which then cap D gives nothing since 11 is not in D. (iv) shared element of A and C. (v) B and D share nothing. (vi) B union C first, then intersect with A. (vii) A and D share nothing. (viii) B union D first, then intersect with A. (ix) A cap B first, then intersect with B union C. (x) A union D and B union C computed separately, then intersected.

   ✦ Working through each part gives: (i) {7,9,11} (ii) {11,13} (iii) phi (iv) {11} (v) phi (vi) {7,9,11} (vii) phi (viii) {7,9,11} (ix) {7,9,11} (x) {7,9,11,15}.

7. 1.4.72 marksNCERT Class 11 Mathematics, Sets, Reprint 2026-27

   If A=naturals, B=even naturals, C=odd naturals, D=primes, find (i) A cap B (ii) A cap C (iii) A cap D (iv) B cap C (v) B cap D (vi) C cap D.

   Hint. Since B and C are both subsets of A, intersecting either with A changes nothing -- the interesting cases are (iv), (v), (vi).

   (i)-(iii) A contains B, C and D entirely, so each intersection is just the smaller set. (iv) no number is both even and odd, so this is empty. (v) the only even prime is 2, so this intersection has one element. (vi) every prime except 2 is odd, so this is D with 2 removed.

   ✦ Working through each part gives: (i) A cap B = B (ii) A cap C = C (iii) A cap D = D (iv) B cap C = phi (v) B cap D = {2} (vi) C cap D = D minus {2}, the set of all odd primes.

8. 1.4.82 marksNCERT Class 11 Mathematics, Sets, Reprint 2026-27

   Which of the following pairs of sets are disjoint? (i) {1,2,3,4} and {x : natural, 4<=x<=6} (ii) {a,e,i,o,u} and {c,d,e,f} (iii) {even integers} and {odd integers}.

   Hint. Two sets are disjoint only if they share absolutely no element.

   (i) both contain 4, so they are not disjoint. (ii) both contain e, so they are not disjoint. (iii) no integer is simultaneously even and odd, so these are disjoint.

   ✦ Working through each part gives: disjoint: (iii) only. Not disjoint: (i) and (ii), since they share an element.

9. 1.4.93 marksNCERT Class 11 Mathematics, Sets, Reprint 2026-27

   If A={3,6,9,12,15,18,21}, B={4,8,12,16,20}, C={2,4,6,8,10,12,14,16}, D={5,10,15,20}, find (i) A-B (ii) A-C (iii) A-D (iv) B-A (v) C-A (vi) D-A (vii) B-C (viii) B-D (ix) C-B (x) D-B (xi) C-D (xii) D-C.

   Hint. A-B keeps only A's elements that are absent from B -- work through each pair carefully in the stated order, since A-B and B-A are rarely the same.

   Remove from the first-named set every element it shares with the second-named set, keeping the rest exactly as given.

   ✦ Working through each part gives: (i) A-B={3,6,9,15,18,21} (ii) A-C={3,9,15,18,21} (iii) A-D={3,6,9,12,18,21} (iv) B-A={4,8,16,20} (v) C-A={2,4,8,10,14,16} (vi) D-A={5,10,20} (vii) B-C={20} (viii) B-D={4,8,12,16} (ix) C-B={2,6,10,14} (x) D-B={5,10,15} (xi) C-D={2,4,6,8,12,14,16} (xii) D-C={5,15,20}.

10. 1.4.102 marksNCERT Class 11 Mathematics, Sets, Reprint 2026-27

    If X={a,b,c,d} and Y={f,b,d,g}, find (i) X-Y (ii) Y-X (iii) X cap Y.

    Hint. Compare the two lists directly, element by element.

    X keeps a and c after removing what it shares with Y (b,d); Y keeps f and g after removing what it shares with X; the shared elements themselves form the intersection.

    ✦ Working through each part gives: (i) X-Y={a,c} (ii) Y-X={f,g} (iii) X cap Y={b,d}.

11. 1.4.111 markNCERT Class 11 Mathematics, Sets, Reprint 2026-27

    If R is the set of real numbers and Q is the set of rational numbers, what is R-Q?

    Hint. Removing every rational number from the reals leaves exactly the numbers that cannot be written as a fraction of integers.

    By definition, a real number that is not rational is irrational, so this difference is precisely the set of irrational numbers.

    ✦ Working through each part gives: r-Q is the set of irrational numbers, sometimes written T.

12. 1.4.122 marksNCERT Class 11 Mathematics, Sets, Reprint 2026-27

    State true or false, with justification: (i) {2,3,4,5} and {3,6} are disjoint. (ii) {a,e,i,o,u} and {a,b,c,d} are disjoint. (iii) {2,6,10,14} and {3,7,11,15} are disjoint. (iv) {2,6,10} and {3,7,11} are disjoint.

    Hint. Check for even a single shared element -- one is enough to make a pair NOT disjoint.

    (i) both sets contain 3, so they are not disjoint. (ii) both sets contain a, so they are not disjoint. (iii) one list is all numbers of the form 4k+2, the other all of the form 4k+3 -- no overlap is possible. (iv) similarly, one list is all multiples of... checking directly, no number appears in both.

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

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.