Exercise 1.4 answers: Sets

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(i) X = {1, 3, 5} Y = {1, 2, 3}

1. Put every member of X and Y together, writing shared members once.
2. 1 and 3 appear in both, so each goes in once.

AnswerX ∪ Y = {1, 2, 3, 5}

(ii) A = [ a, e, i, o, u}

1. A holds the five vowels; B holds a, b, c.
2. a is in both, so it appears once in the union.

AnswerA ∪ B = {a, b, c, e, i, o, u}

(iii) x is a natural number and multiple of 3

1. A is the multiples of 3: 3, 6, 9, 12, and so on.
2. B is the naturals below 6: 1, 2, 3, 4, 5.
3. The union keeps everything below 6, together with every multiple of 3 beyond that.

AnswerA ∪ B = {1, 2, 3, 4, 5, 6, 9, 12, 15, …} − every natural number that is less than 6, or is a multiple of 3

(iv) x is a natural number and 1 < x ≤ 6

1. A = {2,3,4,5,6}; B = {7,8,9}.
2. The two sets share nothing, so the union simply puts them together.

AnswerA ∪ B = {2, 3, 4, 5, 6, 7, 8, 9}

(v) A = {1, 2, 3}, B = φ

1. The empty set adds nothing to a union.

AnswerA ∪ B = {1, 2, 3}

Watch this explained “Why four and four gave six”, 1:44 into Union and intersection, and what it means for two sets to miss each other entirely

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1. Every member of A (a and b) is also a member of B, so A ⊂ B.
2. Since A ⊂ B, their union is just the larger set B.

AnswerA ⊂ B is true; A ∪ B = {a, b, c}

Watch this explained “When one set holds the other”, 2:43 into Union and intersection, and what it means for two sets to miss each other entirely

Open NCERT p. 17Matches NCERT’s answer

1. Every member of A is already a member of B, since A ⊂ B.
2. So putting A and B together adds nothing beyond what B already has.

AnswerA ∪ B = B

Watch this explained “When one set holds the other”, 2:43 into Union and intersection, and what it means for two sets to miss each other entirely

Open NCERT p. 17Checked by computer

(i) A ∪ B

1. A = {1, 2, 3, 4} and B = {3, 4, 5, 6} share 3 and 4, so each is written once.

AnswerA ∪ B = {1, 2, 3, 4, 5, 6}

(ii) A ∪ C

1. A = {1, 2, 3, 4} and C = {5, 6, 7, 8} share nothing, so all eight numbers go in.

AnswerA ∪ C = {1, 2, 3, 4, 5, 6, 7, 8}

(iii) B ∪ C

1. B = {3, 4, 5, 6} and C = {5, 6, 7, 8} share 5 and 6, so each is written once.

AnswerB ∪ C = {3, 4, 5, 6, 7, 8}

(iv) B ∪ D

1. B = {3, 4, 5, 6} and D = {7, 8, 9, 10} share nothing, so all eight numbers go in.

AnswerB ∪ D = {3, 4, 5, 6, 7, 8, 9, 10}

(v) A ∪ B ∪ C

1. From (i), A ∪ B = {1, 2, 3, 4, 5, 6}.
2. C = {5, 6, 7, 8} adds the new members 7 and 8.

AnswerA ∪ B ∪ C = {1, 2, 3, 4, 5, 6, 7, 8}

(vi) A ∪ B ∪ D

1. From (i), A ∪ B = {1, 2, 3, 4, 5, 6}.
2. D = {7, 8, 9, 10} adds 7, 8, 9 and 10, none of which is already there.

AnswerA ∪ B ∪ D = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

(vii) B ∪ C ∪ D

1. From (iii), B ∪ C = {3, 4, 5, 6, 7, 8}.
2. D = {7, 8, 9, 10} adds the new members 9 and 10.

AnswerB ∪ C ∪ D = {3, 4, 5, 6, 7, 8, 9, 10}

Watch this explained “Why four and four gave six”, 1:44 into Union and intersection, and what it means for two sets to miss each other entirely

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(i)

1. The members common to both X and Y are 1 and 3.

AnswerX ∩ Y = {1, 3}

(ii)

1. The only vowel that is also in {a,b,c} is a.

AnswerA ∩ B = {a}

(iii)

1. B only goes up to 5, and the only multiple of 3 up to 5 is 3.

AnswerA ∩ B = {3}

(iv)

1. A = {2,3,4,5,6} and B = {7,8,9} share nothing.

AnswerA ∩ B = φ

(v)

1. Nothing can be common with the empty set.

AnswerA ∩ B = φ

Watch this explained “The and-question”, 5:53 into Union and intersection, and what it means for two sets to miss each other entirely

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(i) A ∩ B

1. A ∩ B keeps only the numbers common to both lists.
2. A has 3,5,7,9,11 and B has 7,9,11,13 — the shared ones are 7, 9, 11.

Answer{7, 9, 11}

(ii) B ∩ C

1. B ∩ C keeps numbers in both B and C.
2. B has 7,9,11,13 and C has 11,13,15 — the shared ones are 11, 13.

Answer{11, 13}

(iii) A ∩ C ∩ D

1. First find A ∩ C: only 11 is common to A and C.
2. Now meet {11} with D = {15, 17}. 11 is not in D, so nothing survives.

Answerthe empty set

(iv) A ∩ C

1. Compare A = {3,5,7,9,11} with C = {11,13,15}.
2. Only 11 appears in both.

Answer{11}

(v) B ∩ D

1. Compare B = {7,9,11,13} with D = {15,17}.
2. No number is in both, so the intersection is empty.

Answerthe empty set

(vi) A ∩ (B ∪ C)

1. First build B ∪ C = {7,9,11,13,15}.
2. Now keep only the members of A that are also in this union: 7, 9, 11.

Answer{7, 9, 11}

(vii) A ∩ D

1. Compare A = {3,5,7,9,11} with D = {15,17}.
2. Nothing is shared.

Answerthe empty set

(viii) A ∩ (B ∪ D)

1. First build B ∪ D = {7,9,11,13,15,17}.
2. Keep only the members of A that are also here: 7, 9, 11.

Answer{7, 9, 11}

(ix) (A ∩ B) ∩ (B ∪ C)

1. A ∩ B = {7,9,11}.
2. B ∪ C = {7,9,11,13,15}.
3. Meeting these two gives {7, 9, 11}, since all three are in both.

Answer{7, 9, 11}

(x) (A ∪ D) ∩ (B ∪ C)

1. A ∪ D = {3,5,7,9,11,15,17}.
2. B ∪ C = {7,9,11,13,15}.
3. The numbers common to both lists are 7, 9, 11, 15.

Answer{7, 9, 11, 15}

Watch this explained “Reading a long expression”, 11:23 into Union and intersection, and what it means for two sets to miss each other entirely

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(i) A ∩ B

1. A is every natural number, so A ∩ B keeps only what B already demands.
2. Every even natural number is itself a natural number, so nothing is thrown away.
3. A ∩ B is exactly B, the even natural numbers.

Answerthe set of even natural numbers, {2, 4, 6, 8, ...}

(ii) A ∩ C

1. Same idea: C, the odd natural numbers, already sits inside A.
2. A ∩ C is exactly C.

Answerthe set of odd natural numbers, {1, 3, 5, 7, ...}

(iii) A ∩ D

1. D, the prime numbers, are all natural numbers too, so D ⊂ A.
2. A ∩ D is exactly D.

Answerthe set of prime numbers, {2, 3, 5, 7, 11, ...}

(iv) B ∩ C

1. B ∩ C would need a natural number that is both even and odd.
2. When a natural number is divided by 2 the remainder is 0 (even) or 1 (odd), never both, so no number qualifies.

Answerthe empty set, φ

(v) B ∩ D

1. B ∩ D keeps the even numbers that are prime.
2. 2 is even and prime. Every other even number 4, 6, 8, ... has 2 as a divisor besides 1 and itself, so it is not prime.
3. So 2 is the only even prime.

Answer{2}

(vi) C ∩ D

1. C ∩ D keeps the prime numbers that are odd.
2. The only prime that is not odd is 2, so this is every prime except 2.

Answerthe set of odd prime numbers, {3, 5, 7, 11, 13, ...}

Watch this explained “Four families that never end”, 12:36 into Union and intersection, and what it means for two sets to miss each other entirely

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(i) {1, 2, 3, 4} and {x : … ≤ x ≤ 6}

1. {x : natural number, 4 ≤ x ≤ 6} is {4, 5, 6}.
2. {1,2,3,4} and {4,5,6} both contain 4, so they share a member.

Answernot disjoint

(ii) { a, e, i, o, u } and { c, d, …}

1. {a,e,i,o,u} and {c,d,e,f} both contain the letter e.

Answernot disjoint

(iii) {x : x is an even integer} and {x : …}

1. No whole number is both even and odd at once.
2. So the two sets share nothing at all.

Answerdisjoint

Watch this explained “When the and-question never passes”, 7:24 into Union and intersection, and what it means for two sets to miss each other entirely

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(i) A – B

1. A – B keeps A's members that B does NOT have.
2. A = {3,6,9,12,15,18,21}, and B has 12 in it, so 12 leaves.
3. What remains is 3, 6, 9, 15, 18, 21.

Answer{3, 6, 9, 15, 18, 21}

(ii) A – C

1. Remove from A any number that is in C.
2. A and C share only 6 and 12.
3. What remains is 3, 9, 15, 18, 21.

Answer{3, 9, 15, 18, 21}

(iii) A – D

1. Remove from A any number that is in D.
2. A and D share only 15.
3. What remains is 3, 6, 9, 12, 18, 21.

Answer{3, 6, 9, 12, 18, 21}

(iv) B – A

1. Remove from B any number that is in A.
2. B and A share only 12.
3. What remains of B is 4, 8, 16, 20.

Answer{4, 8, 16, 20}

(v) C – A

1. Remove from C any number that is in A.
2. C and A share only 6 and 12.
3. What remains of C is 2, 4, 8, 10, 14, 16.

Answer{2, 4, 8, 10, 14, 16}

(vi) D – A

1. Remove from D any number that is in A.
2. D and A share only 15.
3. What remains of D is 5, 10, 20.

Answer{5, 10, 20}

(vii) B – C

1. Remove from B any number that is in C.
2. B = {4,8,12,16,20}; C holds 4, 8, 12 and 16 too.
3. Only 20 is left.

Answer{20}

(viii) B – D

1. Remove from B any number that is in D.
2. B and D share only 20.
3. What remains of B is 4, 8, 12, 16.

Answer{4, 8, 12, 16}

(ix) C – B

1. Remove from C any number that is in B.
2. C = {2,4,6,8,10,12,14,16}; B holds 4, 8, 12, 16 too.
3. What remains is 2, 6, 10, 14.

Answer{2, 6, 10, 14}

(x) D – B

1. Remove from D any number that is in B.
2. D and B share only 20.
3. What remains of D is 5, 10, 15.

Answer{5, 10, 15}

(xi) C – D

1. Remove from C any number that is in D.
2. C and D share only 10.
3. What remains of C is 2, 4, 6, 8, 12, 14, 16.

Answer{2, 4, 6, 8, 12, 14, 16}

(xii) D – C

1. Remove from D any number that is in C.
2. D and C share only 10.
3. What remains of D is 5, 15, 20.

Answer{5, 15, 20}

Watch this explained “Both ways round”, 0:41 into Difference and complement: the same idea with and without a universal set

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(i) X – Y

1. Drop from X any letter that is also in Y.
2. X = {a,b,c,d}; b and d are also in Y, so they leave.
3. What remains is a, c.

Answer{a, c}

(ii) Y – X

1. Drop from Y any letter that is also in X.
2. Y = {f,b,d,g}; b and d are also in X, so they leave.
3. What remains is f, g.

Answer{f, g}

(iii) X ∩ Y

1. Keep only the letters common to both X and Y.
2. Both lists contain b and d.

Answer{b, d}

Watch this explained “Both ways round”, 0:41 into Difference and complement: the same idea with and without a universal set

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1. R − Q keeps every real number that is not a rational number.
2. A rational number is one that can be written as p/q, where p and q are integers and q ≠ 0.
3. The real numbers that cannot be written this way are exactly the irrational numbers, such as √2 and π.
4. So R − Q is the set of all irrational numbers.

AnswerR − Q is the set of all irrational numbers

Watch this explained “Described by what they are not”, 3:22 into The number systems as a chain of containments, and intervals as pieces of R

Open NCERT p. 18Matches NCERT’s answer

(i) { 2, 3, 4, 5} and { 3, 6} are disjoint sets

1. Both sets contain 3, so they are not disjoint.
2. The statement is false.

AnswerFalse — 3 is common to both

(ii) { a, e, i, o, u } and { a, b, c, d } …

1. Both sets contain a, so they are not disjoint.
2. The statement is false.

AnswerFalse — a is common to both

(iii) { 2, 6, 10, 14 } and { 3, 7, 11, 15} are disjoint sets

1. {2,6,10,14} are all even and {3,7,11,15} are all odd.
2. An even number can never equal an odd number, so nothing is shared.
3. The statement is true.

AnswerTrue — no member is shared

(iv) { 2, 6, 10} and {3, 7, 11} are disjoint sets

1. {2,6,10} are all even and {3,7,11} are all odd, so nothing is shared.
2. The statement is true.

AnswerTrue — no member is shared

Watch this explained “When the and-question never passes”, 7:24 into Union and intersection, and what it means for two sets to miss each other entirely