SETS - Q&A

EXERCISE 6(A)

(i) {x : x ∈ Z and x² - 4x = 0} is equal to:

Solution:
Given equation: x² - 4x = 0
x(x - 4) = 0
So, x = 0 or x = 4.
Since x ∈ Z (integers), both 0 and 4 are valid.
The set is {0, 4}.
Answer: (b) {0, 4}

(ii) {x : x ∈ Z and x² = 9} is equal to:

Solution:
Given equation: x² = 9
x = √9 = ±3
So, x = 3 or x = -3.
Since x ∈ Z, both are valid.
The set is {-3, 3}.
Answer: (d) {-3, 3}

(iii) If set A = {x : x ∈ N, x = n - 3, n ∈ N and n < 3}, then set A is:

Solution:
n ∈ N and n < 3 means n can be 1 or 2.
If n = 1, x = 1 - 3 = -2.
If n = 2, x = 2 - 3 = -1.
However, the condition says x ∈ N (Natural numbers). Neither -2 nor -1 are natural numbers.
Therefore, there are no valid elements in set A.
Set A is an empty set { }.
Answer: (b) { }

(iv) Set {x : x = n² - 1, x ∈ Z and 2 < n ≤ 5} is equal to:

Solution:
Since 2 < n ≤ 5, n can take values 3, 4, 5.
When n = 3, x = 3² - 1 = 9 - 1 = 8.
When n = 4, x = 4² - 1 = 16 - 1 = 15.
When n = 5, x = 5² - 1 = 25 - 1 = 24.
The set is {8, 15, 24}.
Answer: (a) {8, 15, 24}

(v) Set-builder form of set A = {1/2, 2/3, 3/4} is:

Solution:
The elements are fractions where the denominator is 1 greater than the numerator.
General term: n / (n + 1).
For 1/2, n=1. For 2/3, n=2. For 3/4, n=3.
So n ∈ N and 1 ≤ n < 4 (or n ≤ 3).
Checking options:
(d) {x = n/(n+1), n ∈ N and 1 ≤ n < 4} matches exactly.
Answer: (d) {x = n/(n+1), n ∈ N and 1 ≤ n < 4}

(i) A₁ = {x : 2x + 3 = 11}

2x + 3 = 11 ⇒ 2x = 8 ⇒ x = 4.
A₁ = {4}

(ii) A₂ = {x : x² - 4x - 5 = 0}

x² - 5x + x - 5 = 0 ⇒ x(x - 5) + 1(x - 5) = 0 ⇒ (x - 5)(x + 1) = 0.
x = 5, -1.
A₂ = {5, -1}

(iii) A₃ = {x : x ∈ Z, -3 ≤ x < 4}

Integers from -3 up to (but not including) 4.
A₃ = {-3, -2, -1, 0, 1, 2, 3}

(iv) A = {x : x is a two digit number and sum of the digits of x is 7}

Possible pairs adding to 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1), (7,0).
A = {16, 25, 34, 43, 52, 61, 70}

(v) A₅ = {x : x = 4n, n ∈ W and n < 4}

n ∈ W and n < 4 means n = 0, 1, 2, 3.
x = 4(0)=0; 4(1)=4; 4(2)=8; 4(3)=12.
A₅ = {0, 4, 8, 12}

(vi) A₆ = {x : x = n / (n+2), n ∈ N and n > 5}

n ∈ N and n > 5 means n = 6, 7, 8, 9, ...
x values: 6/8, 7/9, 8/10, ...
Simplified: 3/4, 7/9, 4/5, ...
A₆ = {3/4, 7/9, 4/5, ...}

(i) B₁ = {6, 9, 12, 15, ...}

These are multiples of 3 starting from 6.
B₁ = {x : x = 3n, n ∈ N and n ≥ 2}

(ii) B₂ = {11, 13, 17, 19}

These are prime numbers between 10 and 20.
B₂ = {x : x is a prime number and 10 < x < 20}

(iii) B₃ = {1/3, 3/5, 5/7, 7/9, 9/11, ...}

Numerators are odd numbers (2n-1). Denominators are 2 greater than numerators (2n+1).
B₃ = {x : x = (2n-1)/(2n+1), n ∈ N}

(iv) B₄ = {8, 27, 64, 125, 216}

These are cubes: 2³, 3³, 4³, 5³, 6³.
B₄ = {x : x = n³, n ∈ N and 2 ≤ n ≤ 6}

(v) B₅ = {-5, -4, -3, -2, -1}

Integers from -5 to -1.
B₅ = {x : x ∈ Z and -5 ≤ x ≤ -1}

(vi) B₆ = {..., -6, -3, 0, 3, 6, ...}

Multiples of 3 (integers).
B₆ = {x : x = 3n, n ∈ Z}

(i) Is {1, 2, 4, 16, 64} = {x : x is a factor of 32}? Give reason.

Factors of 32 are 1, 2, 4, 8, 16, 32.
The given set is {1, 2, 4, 16, 64}.
The sets are not equal because 64 is not a factor of 32, and 8 and 32 are missing from the first set.
Answer: No.

(ii) Is {x : x is a factor of 27} ≠ {3, 9, 27, 54}? Give reason.

Factors of 27 are 1, 3, 9, 27.
The second set is {3, 9, 27, 54}.
Since 1 is in the first set but not the second, and 54 is in the second but not the first, they are not equal.
The statement asks if they are "not equal" (≠), which is true.
Answer: Yes, the statement is true (the sets are not equal).

(i) The set of letters in the word 'MEERUT'.

Elements are not repeated.
{M, E, R, U, T}

(ii) The set of letters in the word 'UNIVERSAL'.

{U, N, I, V, E, R, S, A, L}

(iii) A = {x : x = y + 3, y ∈ N and y > 3}.

y ∈ N and y > 3 means y = 4, 5, 6, ...
x = 4+3=7; 5+3=8; ...
A = {7, 8, 9, ...}

(iv) B = {p : p ∈ W and p² < 20}.

p ∈ W (Whole numbers: 0, 1, 2...).
0²=0 < 20 (Yes)
1²=1 < 20 (Yes)
2²=4 < 20 (Yes)
3²=9 < 20 (Yes)
4²=16 < 20 (Yes)
5²=25 > 20 (No)
B = {0, 1, 2, 3, 4}

(v) C = {x : x is a composite number and 5 ≤ x ≤ 21}

Composite numbers (not prime) between 5 and 21 inclusive.
C = {6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21}

(i) {x : x² - 2x - 3 = 0}

x² - 3x + x - 3 = 0 ⇒ (x-3)(x+1) = 0 ⇒ x = 3, -1.
{3, -1}

(ii) {x : x = 2y + 5; y ∈ N and 2 ≤ y < 6}

y can be 2, 3, 4, 5.
y=2 ⇒ x=9; y=3 ⇒ x=11; y=4 ⇒ x=13; y=5 ⇒ x=15.
{9, 11, 13, 15}

(iii) {x : x is a factor of 24}

{1, 2, 3, 4, 6, 8, 12, 24}

(iv) {x : x ∈ Z and x² ≤ 4}

Squares less than or equal to 4 for integers.
(-2)²=4, (-1)²=1, 0²=0, 1²=1, 2²=4.
{-2, -1, 0, 1, 2}

(v) {x : 3x - 2 ≤ 10 and x ∈ N}

3x ≤ 12 ⇒ x ≤ 4. Since x ∈ N.
{1, 2, 3, 4}

(vi) {x : 4 - 2x > -6, x ∈ Z}

-2x > -10 ⇒ 2x < 10 ⇒ x < 5.
Since x ∈ Z, it implies all integers less than 5.
{..., 2, 3, 4}

(i) Set of integers and set of natural numbers.

Both have unlimited elements.
Answer: Infinite

(ii) Set of whole numbers and set of multiples of 3.

Both are never ending.
Answer: Infinite

(iii) A = {x : x ∈ N and 11 ≥ 2x - 1} and B = {y : y ∈ W and 3 ≤ y ≤ 9}.

For A: 12 ≥ 2x ⇒ 6 ≥ x. x ∈ N. Elements: {1,2,3,4,5,6}. (Finite)
For B: y ∈ W, 3 to 9. Elements: {3,4,5,6,7,8,9}. (Finite)
Answer: Finite

(iv) P = {5, 6, 7, 8} and M = {x : x ∈ W and x ≤ 4}.

P has 4 elements. M = {0, 1, 2, 3, 4}.
Answer: Finite

(i) The set of points of intersection of two non-parallel straight lines on the same plane.

Two non-parallel lines intersect at exactly one point.
Answer: Singleton Set

(ii) A = {x : 7x - 3 = 11}

7x = 14 ⇒ x = 2. Element is {2}.
Answer: Singleton Set

(iii) B = {y : 2y + 1 < 3 and y ∈ W}

2y < 2 ⇒ y < 1. y ∈ W implies y = 0. Set is {0}.
Answer: Singleton Set

(i) The set of points of intersection of two parallel lines.

Parallel lines never intersect.
Answer: Empty Set

(ii) A = {x : x ∈ N and 5 < x ≤ 6}.

x is a natural number strictly greater than 5 and less than or equal to 6. x = 6.
Set is {6}. Not empty.
Answer: Not Empty

(iii) B = {x : x² + 4 = 0 and x ∈ N}.

x² = -4. No natural number squared is negative.
Answer: Empty Set

(iv) C = {even numbers between 6 and 10}.

Even number between 6 and 10 is 8. Set is {8}.
Answer: Not Empty

(v) D = {prime numbers between 7 and 11}.

Primes between 7 and 11: none. (8, 9, 10 are composite).
Answer: Empty Set

Solve B: x² - 6x + x - 6 = 0 ⇒ (x-6)(x+1)=0 ⇒ B = {6, -1}.
A = {4, 5, 6}.
Common element is 6. So they are not disjoint.
Answer: No, they are joint sets.

(ii) Are the sets A = {b, c, d, e} and B = {x : x is a letter in the word 'MASTER'} joint?

B = {M, A, S, T, E, R}.
Common element: 'e' is in both.
Answer: Yes, they are joint sets.

(i) A = {x : x ∈ Z, x < 10} ... [Note: Question text incomplete in image, assuming standard comparison or just set definition needed, but based on (ii) and (iii) being absent in source, solving available parts]

*Note: The scanned page cuts off parts of Q11 (i), (ii), (iii). I will answer the parts visible or inferable.*
(i) A is infinite. If compared to a finite set, not equivalent.

(ii) A = {x : x ∈ W, 5x - 3 ≤ 20} and B = {y : y ∈ W, y ≤ 4} [Inferred from source 974]

Solve A: 5x ≤ 23 ⇒ x ≤ 4.6. x ∈ W ⇒ A = {0, 1, 2, 3, 4}. n(A) = 5.
Solve B: y ∈ W, y ≤ 4 ⇒ B = {0, 1, 2, 3, 4}. n(B) = 5.
Both have 5 elements.
Answer: Equivalent

(i) A = {2, 4, 6, 8} and B = {2n : n ∈ N and n < 5}

B: n = 1, 2, 3, 4. 2n = 2, 4, 6, 8. B = {2, 4, 6, 8}.
Elements are identical.
Answer: Equal

(ii) M = {x : x ∈ W and x + 3 < 8} and N = {y : y = 2n - 1, n ∈ N and n < 5}

M: x < 5, x ∈ W ⇒ {0, 1, 2, 3, 4}.
N: n = 1, 2, 3, 4. y = 1, 3, 5, 7. N = {1, 3, 5, 7}.
Elements are different.
Answer: Not Equal

(iii) E = {x : x² + 8x - 9 = 0} and F = {1, -9}

E: (x+9)(x-1) = 0 ⇒ x = -9, 1. E = {1, -9}.
Elements are identical.
Answer: Equal

(iv) A = {x : x ∈ N, x < 3} and B = {y : y² - 3y + 2 = 0}

A = {1, 2}.
B: (y-2)(y-1) = 0 ⇒ y = 1, 2. B = {1, 2}.
Answer: Equal

(i) The set of multiples of 8.

8, 16, 24... goes on forever.
Answer: Infinite

(ii) The set of integers less than 10.

..., -2, -1, 0, ... 9. Endless in negative direction.
Answer: Infinite

(iii) The set of whole numbers less than 12.

{0, 1, 2, ... 11}. Countable.
Answer: Finite

(iv) {x : x = 3n - 2, n ∈ W, n ≤ 8}

n takes limited values 0 to 8.
Answer: Finite

(v) {x : x = 3n - 2, n ∈ Z, n ≤ 8}

n can be any integer less than or equal to 8 (..., -1, 0, 1...). Infinite values.
Answer: Infinite

(vi) {x : x = (n-2)/(n+1), n ∈ W}

n ∈ W is an infinite set {0, 1, 2...}, so x will have infinite values.
Answer: Infinite

(i) The set of even natural numbers less than 21 and the set of odd natural numbers less than 21 are equivalent sets.

Even < 21: {2, 4, ..., 20} (10 elements).
Odd < 21: {1, 3, ..., 19} (10 elements).
Since cardinal numbers are same (10), they are equivalent.
Answer: True

(ii) If E = {factors of 16} and F = {factors of 20}, then E = F.

E = {1, 2, 4, 8, 16}.
F = {1, 2, 4, 5, 10, 20}.
Elements are different.
Answer: False

(iii) The set A = {integers less than 20} is a finite set.

It includes ..., -2, -1, 0... up to 19. It is infinite in the negative direction.
Answer: False

(iv) If A = {x : x is an even prime number}, then set A is empty.

2 is an even prime number. A = {2}. It is not empty.
Answer: False

(v) The set of odd prime numbers is the empty set.

3, 5, 7... are odd primes. It is not empty.
Answer: False

(vi) The set of squares of integers and the set of whole numbers are equal sets.

Squares: {0, 1, 4, 9, 16...}.
Whole numbers: {0, 1, 2, 3, 4...}.
Numbers like 2, 3 are in Whole numbers but not in Squares.
Answer: False

EXERCISE 6(B)

(i) A set P has 3 elements. The number of proper subsets of set P is:

Solution:
Number of subsets = 2ⁿ = 2³ = 8.
Number of proper subsets = 2ⁿ - 1 = 8 - 1 = 7.
Answer: (d) 7

(ii) For sets A and B, where A = {2, 4, 6} and B = {1, 3, 5, 7}, A ∩ B is:

Solution:
Intersection means common elements.
A has even numbers, B has odd numbers. No common elements.
A ∩ B = Ø.
Answer: (a) Ø

(iii) If set A = {4, 6, 8} and set B = {0}, then A ∪ B:

Solution:
Union combines elements of both sets.
{4, 6, 8, 0}.
Answer: (b) {4, 6, 8, 0}

(iv) If set A = students in class 8 of a particular school and set B = students of this school, then:

Solution:
All students in class 8 (Set A) are also students of the school (Set B).
So, A is a subset of B.
Answer: (c) A ⊂ B

(v) If universal set ξ = {x : x ∈ W, x < 5} and set A = {1, 3}, then complement of set A is equal to:

Solution:
ξ = {0, 1, 2, 3, 4}.
A' (Complement) = ξ - A = {0, 2, 4}.
Answer: (b) {0, 2, 4}

(vi) If universal set = N (set of natural numbers), set A = {multiples of 3 less than or equal to 20} and Set B = {multiples of 4 less than or equal to 20}, then A - B is equal to:

Solution:
A = {3, 6, 9, 12, 15, 18}.
B = {4, 8, 12, 16, 20}.
A - B = Elements in A removing those found in B.
Remove 12 (common).
A - B = {3, 6, 9, 15, 18}.
Answer: (a) {3, 6, 9, 15, 18}

(i) A = {5, 7}

Ø, {5}, {7}, {5, 7}

(ii) B = {a, b, c}

Ø, {a}, {b}, {c}, {a, b}, {b, c}, {a, c}, {a, b, c}

(iii) C = {x : x ∈ W, x ≤ 2}

C = {0, 1, 2}. Subsets:
Ø, {0}, {1}, {2}, {0, 1}, {1, 2}, {0, 2}, {0, 1, 2}

(iv) {p : p is a letter in the word 'poor'}

Unique letters = {p, o, r}. Subsets:
Ø, {p}, {o}, {r}, {p, o}, {o, r}, {p, r}, {p, o, r}

Set C = {c, o, l, e, r}. n(C) = 5.

(iii) number of its subsets

2ⁿ = 2⁵ = 32

(iv) number of its proper subsets

2ⁿ - 1 = 32 - 1 = 31

Set T = {T, E, H}. (Distinct letters).
Subsets:
Ø, {T}, {E}, {H}, {T, E}, {E, H}, {T, H}, {T, E, H}

(i) A = {x : x < 2}

From universal set, select elements < 2.
A = {-7, -3, -1, 0}

(ii) B = {x : -4 < x < 6}

From universal set, select elements between -4 and 6.
B = {-3, -1, 0, 5}

(i) A = {x : x = 3p; p ∈ N}

Multiples of 3 less than 20.
A = {3, 6, 9, 12, 15, 18}

(ii) B = {y : y = 2n + 3, n ∈ N}

n=1 ⇒ 5; n=2 ⇒ 7; ... up to < 20.
B = {5, 7, 9, 11, 13, 15, 17, 19}

(iii) C = {x : x is divisible by 4}

C = {4, 8, 12, 16}

x² - 10x + x - 10 = 0 ⇒ (x-10)(x+1) = 0.
Set = {10, -1}.
Proper subsets exclude the set itself.
Ø, {10}, {-1}

(i) A ⊂ B

False. A (all triangles) is the superset of B (isosceles), not subset.
False

(ii) B ⊂ A

True. Every isosceles triangle is a triangle.
True

(iii) C ⊂ B

True. Every equilateral triangle is isosceles (has at least two equal sides).
True

(iv) B ⊂ A

(Repeated question from ii) True

(v) C ⊂ A

True. Equilateral triangles are triangles.
True

(vi) C ⊂ B ⊂ A

True. Equilateral ⊂ Isosceles ⊂ All Triangles.
True

(i) B ⊂ C

False. Not all rectangles are squares.
False

(ii) D ⊂ B

False. Not all rhombuses are rectangles.
False

(iii) C ⊂ B ⊂ A

True. Squares are Rectangles, Rectangles are Quadrilaterals.
True

(iv) D ⊂ A

True. Rhombuses are Quadrilaterals.
True

(v) B ⊇ C

True. Rectangles contain Squares.
True

(vi) A ⊇ B ⊇ D

False. B (Rectangles) is not a superset of D (Rhombuses).
False

ξ = {10, 11, ..., 35}.
A (within ξ) = {10, 11, 12, 13, 14, 15, 16}.
B (within ξ) = {30, 31, 32, 33, 34, 35}.

(i) A'

A' = ξ - A = {17, 18, ..., 35}.
A' = {17, 18, 19, ..., 35}

(ii) B'

B' = ξ - B = {10, 11, ..., 29}.
B' = {10, 11, ..., 29}

ξ = {-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6}.
N (within ξ) = {0, 1, 2, 3, 4, 5, 6}.
P (within ξ) = {-5, -4, -3, -2, -1, 0}.

(i) N'

N' = ξ - N = {-5, -4, -3, -2, -1}.
N' = {-5, -4, -3, -2, -1}

(ii) P'

P' = ξ - P = {1, 2, 3, 4, 5, 6}.
P' = {1, 2, 3, 4, 5, 6}

M = {R, E, A, L}
N = {L, A, R, E}

(i) M ⊂ N is true.

Every element of M is in N.
True

(ii) N ⊂ M is true.

Every element of N is in M.
True

(iii) M = N is true.

Since M ⊂ N and N ⊂ M, they are equal.
True

A = {4, 5, 6}.
B = {0, 1, 2, 3}.

(i) sets A and B in roster form

A = {4, 5, 6}, B = {0, 1, 2, 3}

(ii) A ∪ B

{0, 1, 2, 3, 4, 5, 6}

(iii) A ∩ B

No common elements.
Ø (Empty set)

(iv) A - B

Elements in A not in B.
{4, 5, 6}

(v) B - A

Elements in B not in A.
{0, 1, 2, 3}

P = {4, 5, 6, 7, 8}.
Q = {1, 2, 3, 4, 5}.

(i) P ∪ Q and P ∩ Q

P ∪ Q = {1, 2, 3, 4, 5, 6, 7, 8}.
P ∩ Q = {4, 5}.

(ii) Is (P ∪ Q) ⊇ (P ∩ Q)?

{1..8} ⊇ {4, 5}. Yes.
Answer: Yes

B = {4, 5, 6, 7}.
C = {1, 2, 3, 4, 5}.

(i) set C

{1, 2, 3, 4, 5}

(ii) n(C)

5

(iii) number of its subsets

2⁵ = 32

(iv) number of its proper subsets

32 - 1 = 31

(i) A ∪ (B ∪ C) = (A ∪ B) ∪ C

B ∪ C = {0, 2, 3, 4, 6, 8, 9}.
A ∪ (B ∪ C) = {0, 1, 2, 3, 4, 5, 6, 8, 9}.

A ∪ B = {0, 1, 2, 4, 5, 6, 8}.
(A ∪ B) ∪ C = {0, 1, 2, 4, 5, 6, 8, 3, 9} = {0, 1, 2, 3, 4, 5, 6, 8, 9}.
Verified.

(ii) A ∩ (B ∩ C) = (A ∩ B) ∩ C

B ∩ C = {0, 6}.
A ∩ (B ∩ C) = {0}.

A ∩ B = {0, 2, 4}.
(A ∩ B) ∩ C = {0}.
Verified.

A = {6, 7, 8, 9}.
B = {3, 4, 5, 6, 7}.
C = {2, 4, 6, 8}.

(i) A ∩ (B ∪ C)

B ∪ C = {2, 3, 4, 5, 6, 7, 8}.
A ∩ (B ∪ C) = {6, 7, 8}.
{6, 7, 8}

(ii) (B ∪ A) ∩ (B ∪ C)

B ∪ A = {3, 4, 5, 6, 7, 8, 9}.
B ∪ C = {2, 3, 4, 5, 6, 7, 8}.
Intersection = {3, 4, 5, 6, 7, 8}.
{3, 4, 5, 6, 7, 8}

(iii) B ∪ (A ∩ C)

A ∩ C = {6, 8}.
B ∪ {6, 8} = {3, 4, 5, 6, 7, 8}.
{3, 4, 5, 6, 7, 8}

(iv) (A ∩ B) ∪ (A ∩ C)

A ∩ B = {6, 7}.
A ∩ C = {6, 8}.
Union = {6, 7, 8}.
{6, 7, 8}

Name the sets which are equal.

Sets from (i) and (iv) are equal. Sets from (ii) and (iii) are equal.

P = {1, 2, 3, 4, 6, 9, 12, 18, 36}.
Q = {1, 2, 3, 4, 6, 8, 12, 16, 24, 48}.

(i) P ∪ Q

{1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48}

(ii) P ∩ Q

{1, 2, 3, 4, 6, 12}

(iii) Q - P

Elements in Q not in P.
{8, 16, 24, 48}

(iv) P' ∩ Q

This is equivalent to Q - P.
{8, 16, 24, 48}

C = {3, 4, 5, 6, 7}.

(i) A - B

{7, 9}
Answer: {7, 9}

(ii) B - C

{8, 10}
Answer: {8, 10}

(iii) B - (A - C)

A - C = {8, 9}.
B - {8, 9} = {4, 6, 10}.
Answer: {4, 6, 10}

(iv) A - (B ∪ C)

B ∪ C = {3, 4, 5, 6, 7, 8, 10}.
A - {3, 4, 5, 6, 7, 8, 10} = {9}.
Answer: {9}

(v) B - (A ∩ C)

A ∩ C = {6, 7}.
B - {6, 7} = {4, 8, 10}.
Answer: {4, 8, 10}

(vi) B - B

Answer: Ø

(i) A - (B ∪ C) = (A - B) ∩ (A - C)

LHS: B ∪ C = {2, 3, 4, 5, 6, 8}. A - (B ∪ C) = {1}.
RHS: A - B = {1, 3, 5}. A - C = {1, 2}. Intersection = {1}.
LHS = RHS. Verified.

(ii) A - (B ∩ C) = (A - B) ∪ (A - C)

LHS: B ∩ C = {4, 6}. A - (B ∩ C) = {1, 2, 3, 5}.
RHS: (A - B) ∪ (A - C) = {1, 3, 5} ∪ {1, 2} = {1, 2, 3, 5}.
LHS = RHS. Verified.

A = {1, 2, 3, 4, 5}.
C: 2x ≤ 13 ⇒ x ≤ 6.5. C = {1, 2, 3, 4, 5, 6}.

(i) A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)

B ∩ C = {3, 6}. A ∪ {3, 6} = {1, 2, 3, 4, 5, 6}.
A ∪ B = {1, 2, 3, 4, 5, 6, 9}.
A ∪ C = {1, 2, 3, 4, 5, 6}.
Intersection = {1, 2, 3, 4, 5, 6}.
Verified.

(ii) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

B ∪ C = {1, 2, 3, 4, 5, 6, 9}. A ∩ (B ∪ C) = {1, 2, 3, 4, 5}.
A ∩ B = {3}.
A ∩ C = {1, 2, 3, 4, 5}.
Union = {1, 2, 3, 4, 5}.
Verified.

Using formula: n(A ∪ B) = n(A) + n(B) - n(A ∩ B)
36 = 30 + 20 - n(A ∩ B)
36 = 50 - n(A ∩ B)
n(A ∩ B) = 50 - 36 = 14.
Answer: 14

n(A ∪ B) = 50 + 30 - 15
n(A ∪ B) = 80 - 15 = 65.
Answer: 65

(i) n(A)

n(A) = n(A - B) + n(A ∩ B) = 30 + 10 = 40.
Answer: 40

(ii) n(B)

n(B) = n(B - A) + n(A ∩ B) = 20 + 10 = 30.
Answer: 30

(iii) n(A ∪ B)

n(A ∪ B) = n(A - B) + n(B - A) + n(A ∩ B) = 30 + 20 + 10 = 60.
Answer: 60

n(A ∪ B) = n(A - B) + n(B - A) + n(A ∩ B)
= 30 + 48 + 15 = 93.
Answer: 93

Test yourself

(i) If A = {3, 5, 7} and B = {5, 7, 9} then n(A ∩ B) is:

A ∩ B = {5, 7}. Count is 2.
Answer: (b) 2

(ii) If n(universal set) = 80 and n(A) = 50 then n(complement of set A) is:

n(A') = 80 - 50 = 30.
Answer: (c) 30

(iii) If A = {5, 8, 10} and empty set Ø. Then A ∪ Ø is equal to:

Union with empty set is the set itself.
Answer: (a) A

(iv) If set A = {x : x ∈ W and 0 < x ≤ 4}, then set A is equal to:

Whole numbers greater than 0 and less than or equal to 4: 1, 2, 3, 4.
Answer: (c) {1, 2, 3, 4}

(v) If set P = {factors of 8}, then set P is equal to:

Factors of 8 are 1, 2, 4, 8.
Answer: (c) {1, 2, 4, 8}

(vi) The elements of the set {x : x ∈ Z and x² ≤ 9} are:

Squares ≤ 9: 0, 1, 4, 9.
Roots: -3, -2, -1, 0, 1, 2, 3.
Answer: (b) {-3, -2, -1, 0, 1, 2, 3} [Note: options order differs, b matches the set]

(vii) If A and B are two equal sets, then A - B is equal to:

Difference of equal sets is empty.
Answer: (d) { }

(viii) If n(A) = n(B) then:

They have same number of elements, so they are equivalent, but not necessarily equal.
However, standard multiple choice logic implies none of the specific equalities hold for sure.
(a) A=B (Not always)
(b) A ≠ B (Not always)
(c) A - B = {0} (No)
Answer: (d) none of these

(ix) A set has 5 elements, then number of its subsets is:

2⁵.
Answer: (a) 2⁵

(x) Let M = {factors of 12} and N = {factors of 24} then {24} is equal to:

M = {1, 2, 3, 4, 6, 12}. N = {1, 2, 3, 4, 6, 8, 12, 24}.
N - M = {8, 24}. This doesn't match {24}.
However, if we look for where 24 exists, it's only in N. {24} ⊂ N.
Let's check options in source (1426): (a) M U N, (b) M n N, (c) M - N, (d) N - M.
N - M = {8, 24}. M - N = { }. M n N = M. M U N = N.
Wait, {24} is not equal to N - M (which is {8, 24}).
Is there a typo in my reading? Source 1420 says "Let M = factors of 12 and N = factors of 24 then {24} is equal to".
Perhaps N - M if we assume 8 is also in M? No, 8 is not factor of 12.
Let's assume the question meant "Which set contains only 24?" or similar.
Actually, if we look strictly: (d) N - M is {8, 24}. None match {24} exactly.
Let's re-read the options. Maybe it asks "factors of 24 but not 12 and not 8"? No.
Given the typical book errors, (d) N - M is the closest "difference" set, though it contains 8 too.
Answer: (d) N - M (Best approximate choice, assuming context of 'elements in N not in M')

(xi) Statement 1: The number of subsets of {{1, {0}}, 2} is 8.
Statement 2: A set containing 'n' elements has 2^n-1 proper subsets.

S1: Set has 2 elements: {1, {0}} and 2. Subsets = 2² = 4. Statement is False.
S2: Proper subsets is 2ⁿ - 1. Statement is False.
Answer: (b) Both the statements are false.

(xii) Assertion (A): Let A = {1, {∅}}, then each of ∅, {1}, {{∅}} is a proper subset of A.
Reason (R): The empty set has no proper subset.

A has elements 1 and {∅}.
Subsets: ∅, {1}, {{∅}}, A.
Proper subsets: ∅, {1}, {{∅}}. So A is True.
R: Empty set has 1 subset (∅), so 0 proper subsets. True.
R does not explain A.
Answer: (b) (2)

(xiii) Assertion (A): Let A = {factors of 12} and B = {factors of 16}. Then B - A = {8, 16}.
Reason (R): B - A = {x | x ∈ A, but x ∉ B}

A = {1, 2, 3, 4, 6, 12}. B = {1, 2, 4, 8, 16}.
B - A = Elements in B not in A = {8, 16}. Assertion is True.
Reason says B - A is x in A not in B. This is the definition of A - B, not B - A. Reason is False.
Answer: (c) (3)

(xiv) Assertion (A): Let A = {1, 2, 3, 4, 5, 6}, and B = {1, 3, 5, 7, 9} then A ∩ B ⊂ A and A ∩ B ⊂ B, always true for every pair of two sets.
Reason (R): For any sets A and B, we have A ∩ B ⊂ A and A ∩ B ⊂ B.

Intersection is always a subset of the original sets. Both True and R explains A.
Answer: (a) (1)

(xv) Assertion (A): Let A = {x | x+3=0, x ∈ N}, B = {x | x ≤ 3, x ∈ W} then A ∩ B = B.
Reason (R): For any set A, A ∩ φ = φ.

A: x = -3. Not in N. A = φ.
B = {0, 1, 2, 3}.
A ∩ B = φ ∩ B = φ.
Assertion says A ∩ B = B. This implies φ = B, which is false (B has elements).
So Assertion is False.
Reason is True.
Answer: (d) (4)

ξ = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}.
A' = {0, 1, 4, 5, 6, 8}.

A = {1, 2, 3, 4, 6, 9, 12, 18, 36}.
B = {1, 2, 3, 4, 6, 8, 12, 16, 24, 48}.

(i) A ∪ B

{1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48}

(ii) A ∩ B

{1, 2, 3, 4, 6, 12}

(iii) A - B

{9, 18, 36}

(iv) B - A

{8, 16, 24, 48}

Let A = {1, 2} and B = {2, 3}.
A - B = {1}, n(A-B)=1.
A ∪ B = {1, 2, 3}, n(A ∪ B) = 3.
n(B) = 2. A ∩ B = {2}, n(A ∩ B) = 1.
n(A) = 2.

(i) n(A - B) = n(A ∪ B) - n(B)

1 = 3 - 2.
1 = 1. Verified.

(ii) n(A ∩ B) + n(A ∪ B) = n(A) + n(B)

1 + 3 = 2 + 2.
4 = 4. Verified.

n(A ∪ B) = n(A - B) + n(B - A) + n(A ∩ B)
= 24 + 32 + 10 = 66.
Answer: 66

ξ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}.
A = {5, 6, 7, 8, 9, 10}.
B = {4, 5}.

(i) (A ∪ B)'

A ∪ B = {4, 5, 6, 7, 8, 9, 10}.
(A ∪ B)' = ξ - {4..10} = {1, 2, 3}.
{1, 2, 3}

(ii) A' ∩ B'

A' = {1, 2, 3, 4}.
B' = {1, 2, 3, 6, 7, 8, 9, 10}.
Intersection = {1, 2, 3}.
{1, 2, 3}

Are (A ∪ B)' and A' ∩ B' equal?

Yes, they are equal (De Morgan's Law).

y = 9, 10, 11, 12.
x = 3(9)-1 = 26.
x = 3(10)-1 = 29.
x = 3(11)-1 = 32.
x = 3(12)-1 = 35.
{26, 29, 32, 35}

ξ = {-2, -1, 0, 1, 2, 3}.
A = {-1, 0, 1, 2}.
B = {1, 2, 3}. (Only those in ξ, so {1, 2, 3})
C = {-2, -1, 0}.

LHS: B ∪ C = {-2, -1, 0, 1, 2, 3}.
A - (B ∪ C) = Ø.

RHS: A - B = {-1, 0}.
A - C = {1, 2}.
Intersection = Ø.
LHS = RHS. Verified.

A: x²=9 ⇒ x=3, -3. A = {3, -3}.
B: x² < 16 ⇒ x < 4. Since x ∈ W, B = {0, 1, 2, 3}.

(i) A ∪ B

{-3, 0, 1, 2, 3}

(ii) B ∩ A

{3}