Where Mindful Learning Begins
Where Mindful Learning Begins
Free, step-by-step Class 11 Maths NCERT Solutions for Chapter 1 Ex 1.4 — all 12 questions solved, with Venn diagrams for union, intersection, set difference, and disjoint sets, for CBSE 2026-27.
This is the longest exercise in the chapter, but it repeats only four operations across many sets: union (combine everything), intersection (keep only the overlap), difference (what's in one set but not the other), and the special case of disjoint sets (no overlap at all). Questions 1–4 build up union, Questions 5–7 do the same for intersection, Question 8 introduces disjoint sets visually, Questions 9–10 cover set difference, and Questions 11–12 close with a classic result (R − Q = irrational numbers) and a set of true/false disjoint checks. Every diagram-worthy idea here is illustrated with a Venn diagram alongside the working.
The union of two sets combines every element that is in either set, writing each shared element only once.
(i) Combine the elements of X and Y, listing 1 and 3 (common to both) only once.
(ii) Combine the elements of A and B; a is common to both and is listed once.
(iii) A=\{3,6,9,12,\ldots\} (multiples of 3), B=\{1,2,3,4,5\}. Combining gives every natural number less than 6, together with every multiple of 3.
(iv) A=\{2,3,4,5,6\}, B=\{7,8,9\}. These sets share no elements, so the union simply lists all of them together.
(v) Since B=\phi has no elements, the union is just A itself.
Both a and b, the elements of A, are also elements of B.
Since A is already entirely contained in B, taking the union adds nothing new — every element of A is already listed in B.
If every element of A already lies in B, then combining A with B introduces no new elements beyond what B already contains.
(i) Combine A and B, listing shared elements 3 and 4 once.
(ii) Combine A and C — they share no elements.
(iii) Combine B and C, listing shared elements 5 and 6 once.
(iv) Combine B and D — they share no elements.
(v) Combine all elements of A, B and C.
(vi) Combine all elements of A, B and D.
(vii) Combine all elements of B, C and D.
The intersection of two sets keeps only the elements common to both.
(i) The elements common to X=\{1,3,5\} and Y=\{1,2,3\} are 1 and 3.
(ii) The only letter common to A=\{a,e,i,o,u\} and B=\{a,b,c\} is a.
(iii) A = multiples of 3, B = {1,2,3,4,5}. The only number that is both a multiple of 3 and less than 6 is 3.
(iv) A=\{2,3,4,5,6\} and B=\{7,8,9\} share no elements.
(v) Since B=\phi has no elements, there can be nothing common to A and B.
(i) Elements common to A and B: 7, 9, 11.
(ii) Elements common to B and C: 11, 13.
(iii) A\cap C=\{11\} (only 11 is common to A and C). Then \{11\}\cap D=\{11\}\cap\{15,17\}=\phi (11 is not in D).
(iv) Elements common to A and C: only 11.
(v) B and D share no elements.
(vi) First, B\cup C=\{7,9,11,13,15\}. Then intersect with A: elements common to \{3,5,7,9,11\} and \{7,9,11,13,15\} are 7, 9, 11.
(vii) A and D share no elements.
(viii) First, B\cup D=\{7,9,11,13,15,17\}. Then intersect with A: common elements are 7, 9, 11.
(ix) A\cap B=\{7,9,11\} from part (i). B\cup C=\{7,9,11,13,15\} from part (vi). Their intersection: 7, 9, 11 (all three already lie in the smaller set).
(x) A\cup D=\{3,5,7,9,11,15,17\}. B\cup C=\{7,9,11,13,15\} from part (vi). Elements common to both: 7, 9, 11, 15.
Note that B\subset A, C\subset A and D\subset A, since every even number, odd number, and prime number is itself a natural number.
(i) Since B is entirely contained in A, their intersection is just B.
(ii) Since C is entirely contained in A, their intersection is just C.
(iii) Since D is entirely contained in A, their intersection is just D.
(iv) No natural number is both even and odd at once.
(v) The only even prime number is 2 — every other prime is odd.
(vi) Every prime number except 2 is odd, so the odd primes are all of D except 2.
Two sets are disjoint only if they share absolutely no elements — their intersection must be empty.
(i) The second set is \{4,5,6\}. It shares the element 4 with \{1,2,3,4\}.
(ii) Both sets contain the letter e.
(iii) No integer can be both even and odd — the two sets share nothing.
A-B keeps only the elements of A that are NOT in B — it removes the overlap but keeps the rest of A.
Working through each part by directly comparing the four given sets:
(i) Remove from A any element also in B (12 is in both).
(ii) Remove from A any element also in C (6 and 12 are in both).
(iii) Remove from A any element also in D (15 is in both).
(iv) Remove from B any element also in A (12 is in both).
(v) Remove from C any element also in A (6 and 12 are in both).
(vi) Remove from D any element also in A (15 is in both).
(vii) Remove from B any element also in C (4, 8, 12, 16 are in both).
(viii) Remove from B any element also in D (20 is in both).
(ix) Remove from C any element also in B (4, 8, 12, 16 are in both).
(x) Remove from D any element also in B (20 is in both).
(xi) Remove from C any element also in D (10 is in both).
(xii) Remove from D any element also in C (10 is in both).
X and Y share the elements b and d.
(i) Remove from X any element also in Y (b and d are in both).
(ii) Remove from Y any element also in X (b and d are in both).
(iii) The elements common to both X and Y.
\mathbb{R}-\mathbb{Q} consists of every real number that is NOT rational.
(i) The two sets share the element 3.
(ii) The two sets share the element a.
(iii) No number appears in both lists.
(iv) No number appears in both lists.
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