Chapter 3 Review Exercise Solutions

Question 1 Multiple Choice Questions

i. The set builder form of the set {1, 1/2, 1/3, 1/4, ...} is:

a) {x | x = n, n ∈ W}

b) {x | x = 1/(2n+1), n ∈ W}

c) {x | x = 1/n, n ∈ N}

d) {x | x = 1/(2n), n ∈ N}

Correct: C. The elements represent reciprocals of consecutive natural numbers starting from 1.

ii. If A = { }, then its power set P(A) is:

a) { }

b) {1}

c) {{ }}

d) ∅

Correct: C. The power set of an empty set is a set containing the empty set as its single element.

iii. If U = {1,2,3,4,5}, A = {1,2,3} and B = {3,4,5}, then U - (A ∩ B) is:

a) {1, 2, 4, 5}

b) {2, 3}

c) {1, 3, 4, 5}

d) {1, 2, 3}

Correct: A. A ∩ B = {3}, so U - {3} = {1, 2, 4, 5}.

iv. If A and B are overlapping sets, then n(A - B) is equal to:

a) n(A)

b) n(B)

c) n(A ∩ B)

d) n(A) - n(A ∩ B)

v. If A ⊂ B and B - A ≠ ∅, then n(B - A) is equal to:

a) 0

b) n(B)

c) n(A)

d) n(B) - n(A)

vi. If n(A ∪ B) = 50, n(A) = 30, and n(B) = 35, then n(A ∩ B) is:

a) 23

b) 15

c) 9

d) 40

Correct: B. PIE formula: 30 + 35 - 50 = 15.

vii. If A = {1, 2, 3, 4} and B = {x, y, z}, then the Cartesian product A × B contains exactly how many elements:

a) 13

b) 12

c) 10

d) 6

Correct: B. |A| × |B| = 4 × 3 = 12.

viii. If f(x) = x² - 3x + 2, then the value of f(a+1) is:

a) a + 1

b) a² + 1

c) a² + 2a + 1

d) a² - a

Correct: D. (a+1)² - 3(a+1) + 2 = a² + 2a + 1 - 3a - 3 + 2 = a² - a.

ix. Given that f(x) = 3x + 1. If f(x) = 28, then the value of x is:

a) 9

b) 27

c) 3

d) 18

x. Let A = {1, 2, 3} and B = {a, b}. A function f: A → B is defined as f = {(1, a), (2, b), (3, b)}. Which statement is true?

a) f is injective

b) f is surjective

c) f is bijective

d) f is into only

Correct: B. The range is {a, b} which equals codomain B, making it surjective (onto).

Question 2 Tabular Form

Write each of the following sets in tabular forms:

(i) {x | x = 2n, n ∈ N}

Solution: The set of positive even integers: {2, 4, 6, 8, ...}.

(ii) {x | x = 2m+1, m ∈ N}

Solution: Natural odd numbers starting from m=1: {3, 5, 7, 9, ...}.

(iii) {x | x = 11n, n ∈ W and n < 11}

Solution: Multiples of 11 for whole numbers n=0 to n=10: {0, 11, 22, 33, 44, 55, 66, 77, 88, 99, 110}.

(iv) {x ∈ E | 4 < x < 6}

Solution: There are no even integers strictly between 4 and 6: { } (or ∅).

(v) {x ∈ O | 5 ≤ x < 7} (where O represents odd integers)

Solution: The only odd integer in this range is 5: {5}.

(vi) {x ∈ Q | x² = 2}

Solution: The equation has solutions x = ±√2, which are irrational. Thus, no rational number satisfies this: { }.

(vii) {x ∈ Q | x = -x}

Solution: Solve the equation: 2x = 0 ⇒ x = 0. Since 0 is rational: {0}.

(viii) {x ∈ R | x ∉ Q'} (where Q' represents irrationals)

Solution: Real numbers that are not irrational are rational numbers: Q.

Question 3 List Set Members

Let U = {1, 2, 3, ..., 10}, A = {2, 4, 6, 8, 10}, B = {1, 2, 3, 4, 5}, and C = {1, 3, 5, 7, 9}. List members of:
(i) A' (ii) B' (iii) A ∪ B (iv) A - B (v) A ∩ C (vi) A' ∪ C' (vii) A' ∪ C (viii) U'

Solutions

(i) A' = U - A = {1, 3, 5, 7, 9}

(ii) B' = U - B = {6, 7, 8, 9, 10}

(iii) A ∪ B = {1, 2, 3, 4, 5, 6, 8, 10}

(iv) A - B = {6, 8, 10}

(v) A ∩ C = { } (Disjoint sets)

(vi) A' ∪ C' = {1, 3, 5, 7, 9} ∪ {2, 4, 6, 8, 10} = {1, 2, 3, ..., 10} = U

(vii) A' ∪ C = {1, 3, 5, 7, 9} ∪ {1, 3, 5, 7, 9} = {1, 3, 5, 7, 9} = C

(viii) U' = U - U = { }

Question 4 Set Simplifications

State the single sets equal to the following:
(i) A' ∩ A (ii) A ∩ U (iii) A ∪ U (iv) A ∪ ∅ (v) ∅ ∩ ∅

Solutions

(i) A' ∩ A = ∅ (A set and its complement have no common elements).

(ii) A ∩ U = A (Intersection with Universal set).

(iii) A ∪ U = U (Union with Universal set).

(iv) A ∪ ∅ = A (Union with empty set).

(v) ∅ ∩ ∅ = ∅.

Question 5 & 6 & 7 Algebraic Set Proofs

Proofs of Set laws and De Morgan's properties.

Solutions & Proof Outline

Q5 (i) Verify A - B = A ∩ B':
- LHS represents elements in A but not in B.
- RHS represents elements in A and in the complement of B (not in B). Hence verified.

Q6 (i) Verify Associativity and Distributivity:
Using A = {1, 2, 3, 4}, B = {3, 4, 5, 6, 7, 8}, C = {5, 6, 7, 9, 10}:
- Union Associativity: (A ∪ B) ∪ C = A ∪ (B ∪ C) = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}.
- Distributivity of Union over Intersection: A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) = {1, 2, 3, 4, 5, 6, 7}.

Q7 De Morgan's verification:
Under U={1, ..., 20}, A=even, B=odd. Since A and B partition U:
- A ∪ B = U ⇒ (A ∪ B)' = ∅.
- A' ∩ B' = B ∩ A = ∅ (as B is even-complements and A is odd-complements). Verified.

Question 8 Intersection of Set Multiples

Consider P = {x | x = 5m, m ∈ N} and Q = {x | x = 2m, m ∈ N}. Find P ∩ Q.

Solution

Set P consists of multiples of 5: P = {5, 10, 15, 20, 25, 30, ...}.

Set Q consists of multiples of 2: Q = {2, 4, 6, 8, 10, 12, ..., 20, ...}.

The intersection P ∩ Q contains common multiples of 5 and 2, which are multiples of 10 (since LCM of 2 and 5 is 10):
P ∩ Q = {10, 20, 30, 40, ...} = {x | x = 10m, m ∈ N}.

Question 9 Absorption Laws

Deduce from properties: (i) A ∩ (A ∪ B) = A ∪ (A ∩ B).

Solution

LHS: A ∩ (A ∪ B) = (A ∩ A) ∪ (A ∩ B) (Distributive law)
= A ∪ (A ∩ B) (Idempotent law: A ∩ A = A).

This matches RHS. Both sides simplify directly to A by the Absorption Laws.

Question 10 Function Calculations

If g(x) = 7x - 2 and s(x) = 8x² - 3, find:
(i) g(0) (ii) g(-1) (iii) g(-5/3) (iv) s(1) (v) s(-9) (vi) s(-7/2)

Solutions

(i) g(0) = 7(0) - 2 = -2.

(ii) g(-1) = 7(-1) - 2 = -7 - 2 = -9.

(iii) g(-5/3) = 7(-5/3) - 2 = -35/3 - 6/3 = -41/3.

(iv) s(1) = 8(1)² - 3 = 8 - 3 = 5.

(v) s(-9) = 8(-9)² - 3 = 8(81) - 3 = 648 - 3 = 645.

(vi) s(-7/2) = 8(-7/2)² - 3 = 8(49/4) - 3 = 2(49) - 3 = 98 - 3 = 95.

Question 11 Solve Parameters

If f(x) = ax + b, f(-2) = 3 and f(4) = 10, find the values of constants a and b.

Solution

Formulate equations:

f(-2) = a(-2) + b = 3 ⇒ -2a + b = 3 (Eq. 1) f(4) = a(4) + b = 10 ⇒ 4a + b = 10 (Eq. 2)

Subtract Eq. 1 from Eq. 2:

(4a + b) - (-2a + b) = 10 - 3 6a = 7 a = 76

Substitute a = 7/6 into Eq. 2:

4(76) + b = 10 143 + b = 10 b = 10 - 143 = 163

Conclusion: a = 7/6 and b = 16/3.

Question 12 Solve x

If k(x) = 7x - 5 and k(x) = 100, find the value of x.

Solution

7x - 5 = 100 7x = 105 x = 15

Question 13 Quadratic constants

If g(x) = mx² + n, g(4) = 20, and g(0) = 5, find m and n.

Solution

Use g(0) = 5:
g(0) = m(0)² + n = 5 ⇒ n = 5.

Use g(4) = 20 with n = 5:

m(4)² + 5 = 20 16m = 15 m = 1516

Conclusion: m = 15/16 and n = 5.

Question 14 Category Unions

Universal set U = products labeled 1 to 100.
- Set A (Electronics): products 1 to 30.
- Set B (Clothing): products 31 to 55.
- Set C (Beauty): products 76 to 100.
Write sets in tabular form, and find their union.

Solution

A = {1, 2, ..., 30}, B = {31, 32, ..., 55}, C = {76, 77, ..., 100}.

Union A ∪ B ∪ C = {1, 2, ..., 55, 76, 77, ..., 100}.

Question 15 Exam Survey

Out of 180 students: 120 passed math, 90 passed science, 60 passed both.
(a) passed either? (b) passed neither? (c) passed science but not math? (d) failed science?

Solution

(a) Passed either: n(M ∪ S) = 120 + 90 - 60 = 150.

(b) Passed neither: 180 - 150 = 30.

(c) Passed science only: n(S) - n(M ∩ S) = 90 - 60 = 30.

(d) Failed science: Total - n(S) = 180 - 90 = 90.

Question 16 Software House PIE

Survey of 300 software developers: 150 like Python (P), 130 like Java (J), 120 like PHP (H).
Double overlaps: 70 like P & J, 50 like J & H, 60 like P & H. 40 developers like all three.
Find: (a) at least one language? (b) only one language? (c) none? (d) only PHP?

Solution

(a) At least one (Union):
n(P ∪ J ∪ H) = 150 + 130 + 120 - 70 - 50 - 60 + 40 = 260.

(b) Only one language:
- Python only: 150 - 70 - 60 + 40 = 60.
- Java only: 130 - 70 - 50 + 40 = 50.
- PHP only: 120 - 50 - 60 + 40 = 50.
Total only one language = 60 + 50 + 50 = 160.

(c) Do not use any: 300 - 260 = 40.

(d) Only PHP: 50.

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Step-by-Step Solution

Chapter 3 Self-Assessment Quiz

9th Mathematics

Review Exercise 3 Solutions