Exercise 1.5 — Sets

1. 1.5.13 marksNCERT Class 11 Mathematics, Sets, Reprint 2026-27

   Let U={1,...,9}, A={1,2,3,4}, B={2,4,6,8}, C={3,4,5,6}. Find (i) A' (ii) B' (iii) (A cup C)' (iv) (A cup B)' (v) (A')' (vi) (B-C)'.

   Hint. The complement of a set is everything in U left over after removing that set's own elements.

   (i) remove A's elements from U. (ii) remove B's elements from U. (iii) find A cup C first, then remove it from U. (iv) find A cup B first, then remove it from U. (v) complementing twice returns the original set. (vi) find B-C first, then remove it from U.

   ✦ Working through each part gives: (i) A'={5,6,7,8,9} (ii) B'={1,3,5,7,9} (iii) (A cup C)'={7,8,9} (iv) (A cup B)'={5,7,9} (v) (A')'=A={1,2,3,4} (vi) (B-C)'={1,3,4,5,6,7,9}.

2. 1.5.22 marksNCERT Class 11 Mathematics, Sets, Reprint 2026-27

   If U={a,b,c,d,e,f,g,h}, find the complements of: (i) A={a,b,c} (ii) B={d,e,f,g} (iii) C={a,c,e,g} (iv) D={f,g,h,a}.

   Hint. List every letter of U that was NOT named in the given set.

   Subtract each set's letters from the full eight-letter universal set.

   ✦ Working through each part gives: (i) A'={d,e,f,g,h} (ii) B'={a,b,c,h} (iii) C'={b,d,f,h} (iv) D'={b,c,d,e}.

3. 1.5.33 marksNCERT Class 11 Mathematics, Sets, Reprint 2026-27

   Taking the naturals as the universal set, write the complements of: (i) even naturals (ii) odd naturals (iii) positive multiples of 3 (iv) primes (v) naturals divisible by both 3 and 5 (vi) perfect squares (vii) perfect cubes (viii) {x:x+5=8} (ix) {x:2x+5=9} (x) {x>=7} (xi) {x in N : 2x+1>10}.

   Hint. Solve any equation or inequality first to identify the actual set before complementing it.

   (i)-(vii) directly negate the stated property. (viii) x+5=8 gives x=3, so the set is {3}. (ix) 2x+5=9 gives x=2, so the set is {2}. (x) the complement of 'at least 7' is 'less than 7'. (xi) 2x+1>10 means x>4.5, i.e. x>=5, so the set is {5,6,7,...}.

   ✦ Working through each part gives: (i) odd naturals (ii) even naturals (iii) naturals not divisible by 3 (iv) composite numbers together with 1 (v) naturals not divisible by both 3 and 5 (vi) naturals that are not perfect squares (vii) naturals that are not perfect cubes (viii) N minus {3} (ix) N minus {2} (x) {1,2,3,4,5,6} (xi) {1,2,3,4}.

4. 1.5.43 marksNCERT Class 11 Mathematics, Sets, Reprint 2026-27

   If U={1,...,9}, A={2,4,6,8}, B={2,3,5,7}, verify (i) (A cup B)' = A' cap B' (ii) (A cap B)' = A' cup B'.

   Hint. Compute each side of both identities completely independently, then compare -- this is exactly what 'verify' requires.

   A cup B={2,3,4,5,6,7,8}, so (A cup B)'={1,9}. Separately A'={1,3,5,7,9}, B'={1,4,6,8,9}, so A' cap B'={1,9} -- both sides match. A cap B={2}, so (A cap B)'={1,3,4,5,6,7,8,9}. Separately A' cup B'={1,3,4,5,6,7,8,9} -- both sides match again.

   ✦ Working through each part gives: both De Morgan identities check out: (A cup B)'=A' cap B'={1,9}, and (A cap B)'=A' cup B'={1,3,4,5,6,7,8,9}.

5. 1.5.52 marksNCERT Class 11 Mathematics, Sets, Reprint 2026-27

   Draw an appropriate Venn diagram for each of: (i) (A cup B)' (ii) A' cap B' (iii) (A cap B)' (iv) A' cup B'.

   Hint. Shade the region OUTSIDE both circles for a union-based complement, and the region outside each individual circle, then combine, for an intersection of complements.

   For (i) and (ii): shade everything in the rectangle except the two circles entirely -- the region outside both A and B. This is the same shaded region for both, since De Morgan's law makes them equal. For (iii) and (iv): shade everything except the small overlap region shared by A and B -- again identical for both, by the second De Morgan law.

   ✦ Working through each part gives: (i) and (ii) shade the same region (everything outside both circles); (iii) and (iv) shade the same region (everything except the overlap) -- visually confirming both De Morgan laws.

6. 1.5.62 marksNCERT Class 11 Mathematics, Sets, Reprint 2026-27

   Let U be the set of all triangles in a plane. If A is the set of triangles with at least one angle different from 60 degrees, what is A'?

   Hint. A' consists of every triangle for which the defining property of A fails completely -- no angle may differ from 60 degrees.

   If not even one angle may differ from 60 degrees, all three angles must equal 60 degrees, which is exactly the condition for an equilateral triangle.

   ✦ Working through each part gives: a' is the set of all equilateral triangles.

7. 1.5.72 marksNCERT Class 11 Mathematics, Sets, Reprint 2026-27

   Fill in the blanks: (i) A cup A' = ... (ii) phi' cap A = ... (iii) A cap A' = ... (iv) U' cap A = ....

   Hint. Recall the complement laws directly: A cup A' always fills the universal set, and A cap A' always empties out.

   (i) a set together with its complement covers everything. (ii) the complement of the empty set is the universal set, and intersecting U with A just gives A back. (iii) a set and its complement share nothing. (iv) the complement of U is the empty set, and intersecting with anything gives the empty set.

   ✦ Working through each part gives: (i) U (ii) A (iii) phi (iv) phi.

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.