Exercise 1.5 answers: Sets

Open NCERT p. 20Matches NCERT’s answer

(i) A′

1. A′ is everything in U that is not in A.
2. U has 1..9; take away 1,2,3,4.
3. What is left is 5, 6, 7, 8, 9.

Answer{5, 6, 7, 8, 9}

(ii) B′

1. Take U and remove B's members 2, 4, 6, 8.
2. What is left is 1, 3, 5, 7, 9.

Answer{1, 3, 5, 7, 9}

(iii) (A ∪ C)′

1. A ∪ C = {1,2,3,4,5,6}.
2. Take U and remove these six numbers.
3. What is left is 7, 8, 9.

Answer{7, 8, 9}

(iv) (A ∪ B)′

1. A ∪ B = {1,2,3,4,6,8}.
2. Take U and remove these numbers.
3. What is left is 5, 7, 9.

Answer{5, 7, 9}

(v) (A′)′

1. A′ = {5,6,7,8,9} from part (i).
2. Complementing again undoes it, giving back A.
3. So (A′)′ = A = {1,2,3,4}.

Answer{1, 2, 3, 4}

(vi) (B – C)′

1. B – C keeps B's members that are not in C: B={2,4,6,8}, C={3,4,5,6}, so 4 and 6 leave.
2. B – C = {2, 8}.
3. Now take U and remove 2 and 8.
4. What is left is 1, 3, 4, 5, 6, 7, 9.

Answer{1, 3, 4, 5, 6, 7, 9}

Watch this explained “Complement is difference, held still”, 5:32 into Difference and complement: the same idea with and without a universal set

Open NCERT p. 20Matches NCERT’s answer

(i) A = {a, b, c}

1. U has a through h.
2. Remove a, b, c.
3. What remains is d, e, f, g, h.

Answer{d, e, f, g, h}

(ii) B = {d, e, f, g}

1. Remove d, e, f, g from U.
2. What remains is a, b, c, h.

Answer{a, b, c, h}

(iii) C = {a, c, e, g}

1. Remove a, c, e, g from U.
2. What remains is b, d, f, h.

Answer{b, d, f, h}

(iv) D = { f, g, h, a}

1. Remove f, g, h, a from U.
2. What remains is b, c, d, e.

Answer{b, c, d, e}

Watch this explained “Complement is difference, held still”, 5:32 into Difference and complement: the same idea with and without a universal set

Open NCERT p. 20Checked by computer

(i) x is an even natural number

1. The complement of "even" among natural numbers is "odd".

Answerthe set of odd natural numbers

(ii) x is an odd natural number

1. The complement of "odd" among natural numbers is "even".

Answerthe set of even natural numbers

(iii) x is a positive multiple of 3

1. The multiples of 3 are 3, 6, 9, ...
2. Everything else — the natural numbers 3 does not divide exactly — is the complement.

Answerthe natural numbers that are not multiples of 3

(iv) x is a prime number

1. The prime numbers are 2, 3, 5, 7, 11, ...
2. The natural numbers that are not prime are 1 (which is neither prime nor composite) and the composite numbers 4, 6, 8, 9, 10, ...

Answerthe natural numbers that are not prime: 1 together with all composite numbers, {1, 4, 6, 8, 9, 10, ...}

(v) x is a natural number divisible by 3 and 5

1. A number divisible by both 3 and 5 is exactly a multiple of 15: 15, 30, 45, ...
2. The complement is every natural number that is not a multiple of 15.

Answerthe natural numbers that are not divisible by both 3 and 5 (not multiples of 15)

(vi) x is a perfect square

1. Perfect squares are 1, 4, 9, 16, ...
2. The complement is every natural number that is not one of these.

Answerthe natural numbers that are not perfect squares

(vii) x is a perfect cube

1. Perfect cubes are 1, 8, 27, 64, ...
2. The complement is every natural number that is not one of these.

Answerthe natural numbers that are not perfect cubes

(viii) x + 5 = 8

1. x + 5 = 8 gives x = 3, so this set is just {3}.
2. The complement is every natural number except 3.

Answer{1, 2, 4, 5, 6, ...} — every natural number except 3

(ix) 2x + 5 = 9

1. 2x + 5 = 9 gives 2x = 4, so x = 2. The set is {2}.
2. The complement is every natural number except 2.

Answer{1, 3, 4, 5, 6, ...} — every natural number except 2

(x) x ≥ 7

1. {x : x ≥ 7} is 7, 8, 9, ...
2. What is left out of the naturals is 1 up to 6.

Answer{1, 2, 3, 4, 5, 6}

(xi) x ∈ N and 2x + 1 > 10

1. 2x + 1 > 10 gives 2x > 9, so x > 4.5.
2. The smallest natural number bigger than 4.5 is 5, so this set is {5, 6, 7, ...}.
3. What is left out is 1, 2, 3, 4.

Answer{1, 2, 3, 4}

Watch this explained “A condition is enough”, 10:39 into Difference and complement: the same idea with and without a universal set

Open NCERT p. 20Checked by computer

1. A ∪ B = {2,3,4,5,6,7,8}, so (A ∪ B)′ = U minus that = {1, 9}.
2. A′ = {1,3,5,7,9} and B′ = {1,4,6,8,9}. Their overlap A′ ∩ B′ is {1, 9} — the same as (A ∪ B)′.
3. A ∩ B = {2}, so (A ∩ B)′ = U minus {2} = {1,3,4,5,6,7,8,9}.
4. A′ ∪ B′ combines {1,3,5,7,9} and {1,4,6,8,9}, giving {1,3,4,5,6,7,8,9} — the same as (A ∩ B)′.

AnswerBoth De Morgan laws check out: (A ∪ B)′ = A′ ∩ B′ = {1, 9}, and (A ∩ B)′ = A′ ∪ B′ = {1, 3, 4, 5, 6, 7, 8, 9}

Watch this explained “Both laws, one pair of sets”, 11:52 into Why complementing turns each of the two operations into the other

Open NCERT p. 20One way to think about it

(i) (A ∪ B)′

1. Draw a rectangle for U with two overlapping circles A and B inside it.
2. A ∪ B is everything inside either circle.
3. Shade the region OUTSIDE both circles — that is (A ∪ B)′.

In shortShade only the part of the rectangle that lies outside both circles.

(ii) A′ ∩ B′

1. A′ is everything outside circle A; B′ is everything outside circle B.
2. A′ ∩ B′ needs both at once, so shade wherever the two outside regions overlap.

In shortShade the part of the rectangle outside both circles — the same region as (i).

(iii) (A ∩ B)′

1. A ∩ B is only the lens where the two circles overlap.
2. Shade everything in the rectangle except that lens.

In shortShade the whole rectangle except the overlapping lens of A and B.

(iv) A′ ∪ B′

1. A′ is outside A; B′ is outside B.
2. A′ ∪ B′ is everything that is in at least one of those two outside regions, which is everywhere except the lens common to both circles.

In shortShade the whole rectangle except the overlapping lens of A and B — the same region as (iii).

Watch this explained “Four shadings, two verdicts”, 8:38 into Turning a claim about sets into a picture you can read off

Open NCERT p. 20Matches NCERT’s answer

1. A′ holds every triangle that is NOT in A.
2. Not being in A means: it is false that some angle differs from 60°, i.e. every angle equals 60°.
3. A triangle with all three angles equal to 60° is exactly an equilateral triangle.

AnswerA′ is the set of all equilateral triangles

Watch this explained “A condition is enough”, 10:39 into Difference and complement: the same idea with and without a universal set

Open NCERT p. 20Matches NCERT’s answer

(i) A ∪ A′ = . . .

1. A set together with everything it is missing (its complement) covers the whole universal set.

AnswerU

(ii) φ′ ∩ A = . . .

1. φ′ is the complement of the empty set, which is the whole universal set U.
2. U ∩ A keeps only what A already has, so this is just A.

AnswerA

(iii) A ∩ A′ = . . .

1. A and A′ share nothing at all — every object is in exactly one of the two.

Answerthe empty set

(iv) U′ ∩ A = . . .

1. U′ is everything outside U, and nothing is outside U — so U′ is the empty set.
2. The empty set meets nothing at all, so the answer is empty.

Answerthe empty set

Watch this explained “Which of the four actually survive”, 8:13 into Difference and complement: the same idea with and without a universal set