Maths Week 1 - Comprehensive Learning Materials

Topics Covered

• Set Theory: Number system, Sets and their operations
• Relations: Relations and their types
• Functions: Functions and their types

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📚 CONCEPTS

1. Number Systems

Natural Numbers (ℕ)

• Set of positive integers: {1, 2, 3, 4, …}
• Used for counting and ordering

Integers (ℤ)

• Set of whole numbers including negatives: {…, -3, -2, -1, 0, 1, 2, 3, …}
• Includes natural numbers, their negatives, and zero

Rational Numbers (ℚ)

• Numbers that can be expressed as p/q where p, q ∈ ℤ and q ≠ 0
• Examples: 1/2, -3/4, 5 (which is 5/1), 0.25 (which is 1/4)
• Terminating decimals: 0.5 = 1/2, 0.75 = 3/4
• Repeating decimals: 0.333… = 1/3, 0.142857… = 1/7

Irrational Numbers

• Numbers that cannot be expressed as p/q where p, q ∈ ℤ
• Non-terminating, non-repeating decimals
• Examples: √2, √3, π, e
• Operations with irrationals:
  • √a × √b = √(ab) when a, b ≥ 0
  • √a + √b is generally irrational unless one term rationalizes the other

Real Numbers (ℝ)

• Union of rational and irrational numbers
• Complete number line with no gaps

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2. Set Theory

Basic Definitions

• Set: Collection of distinct objects
• Element: Member of a set, denoted by ∈
• Cardinality: Number of elements in a set, denoted by |A|

Set Operations

• Union (A ∪ B): All elements in A or B or both
• Intersection (A ∩ B): Elements common to both A and B
• Difference (A \ B): Elements in A but not in B
• Symmetric Difference: Elements in exactly one of the sets

Venn Diagrams

• Visual representation of set relationships
• Useful for solving problems involving multiple sets

Set Properties

• Commutative: A ∪ B = B ∪ A, A ∩ B = B ∩ A
• Associative: (A ∪ B) ∪ C = A ∪ (B ∪ C)
• Distributive: A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

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3. Relations

Definition

A relation R from set A to set B is a subset of A × B

• Written as: R ⊆ A × B
• Ordered pair (a, b) ∈ R means “a is related to b”

Types of Relations

• Reflexive: (a, a) ∈ R for all a ∈ A
• Symmetric: If (a, b) ∈ R, then (b, a) ∈ R
• Transitive: If (a, b) ∈ R and (b, c) ∈ R, then (a, c) ∈ R
• Equivalence Relation: Reflexive, symmetric, and transitive

Relation Properties

• Domain: Set of all first elements
• Range: Set of all second elements
• Inverse Relation: R⁻¹ = {(b, a) | (a, b) ∈ R}

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4. Functions

Definition

A function f: A → B is a relation where:

• Each element in A is related to exactly one element in B
• Domain: Set A
• Codomain: Set B
• Range: Set of actual output values

Types of Functions

• One-to-One (Injective): Different inputs give different outputs
• Onto (Surjective): Every element in codomain has a preimage
• Bijective: Both one-to-one and onto

Function Operations

• Domain of f + g: D_f ∩ D_g
• Composition: (f ∘ g)(x) = f(g(x))
• Inverse Function: f⁻¹ exists if f is bijective

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🎯 QUESTION PATTERNS FROM ANALYSIS

Pattern 1: Irrational Number Identification

Question Type: Determine which expressions are irrational Key Approach: Simplify expressions and check if result can be written as p/q

Pattern 2: Function Domain and Cardinality

Question Type: Find domain restrictions and count excluded integers Key Approach: Identify values making denominator zero or radicand negative

Pattern 3: Set Relations and Cardinality

Question Type: Work with relations defined by mathematical conditions Key Approach: List relation elements systematically, apply set operations

Pattern 4: Venn Diagram Applications

Question Type: Real-world scenarios with overlapping categories Key Approach: Use inclusion-exclusion principle

Pattern 5: Relation Properties

Question Type: Determine if relation is reflexive, symmetric, transitive Key Approach: Test each property systematically

Pattern 6: Function Classification

Question Type: Classify functions as injective, surjective, bijective Key Approach: Check one-to-one and onto conditions

Pattern 7: Complex Set Operations

Question Type: Evaluate expressions with multiple set operations Key Approach: Apply operations step by step, use Venn diagrams

Pattern 8: Real-world Relations

Question Type: Family trees, organizational structures Key Approach: Model relationships mathematically

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📝 PRACTICE EXERCISES

Exercise Set 1: Number Systems

Question 1

Which of the following are irrational numbers?

1. $(\sqrt{12}+\sqrt{3})(\sqrt{8}-\sqrt{2})$
2. $(\sqrt{18}-\sqrt{8})(\sqrt{6}+\sqrt{2})$
3. $\frac{\sqrt{10}+\sqrt{5}}{\sqrt{10}-\sqrt{5}}$
4. $\frac{\sqrt{12}+\sqrt{3}}{\sqrt{12}-\sqrt{3}}$

Question 2

Simplify and determine if rational or irrational:

1. $\sqrt{50}-\sqrt{18}$
2. $(\sqrt{7}+\sqrt{3}{)}^2$
3. $\sqrt{2}\times \sqrt{8}$
4. $\frac{\sqrt{20}}{\sqrt{5}}$

Question 3

Let $a=\sqrt{2}+\sqrt{3}$ and $b=\sqrt{2}-\sqrt{3}$. Find:

1. $a\times b$
2. $a^2+b^2$
3. Determine if $a^2$ and $b^2$ are rational

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Exercise Set 2: Function Domain and Range

Question 4

Find the domain of $f(x)=\frac{\sqrt{x^2-25}}{x-5}$ where $x\in \mathbb{Z}$. If $A$ is the set of integers not in the domain, find $|A|$.

Question 5

Consider $g(x)=\frac{\sqrt{9-x^2}}{x+3}$ where $x\in \mathbb{Z}$. Find the number of integers in the domain.

Question 6

For $h(x)=\frac{\sqrt{x^2-36}}{x+6}$ with $x\in \mathbb{Z}$, let $B$ be the set of excluded integers. Find $|B|$.

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Exercise Set 3: Set Relations

Question 7

Let $S=\{n\mid n\in \mathbb{N},n\le 24\}$. Define relations:

• $R_1=\{(x,y)\mid x,y\in S,y=2x\}$
• $R_2=\{(x,y)\mid x,y\in S,y=x^2\}$

Find $|R_1\setminus (R_1\cap R_2)|$.

Question 8

For $T=\{n\mid n\in \mathbb{N},n\le 20\}$, define:

• $R_1=\{(x,y)\mid x,y\in T,y=3x\}$
• $R_2=\{(x,y)\mid x,y\in T,y=2x\}$

Find $|R_1\cup R_2|$.

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Exercise Set 4: Venn Diagram Applications

Question 9

In a college, 80 students study Mathematics, 70 study Physics, and 60 study Chemistry. 40 study both Mathematics and Physics, 35 study both Mathematics and Chemistry, 30 study both Physics and Chemistry. 20 study all three subjects. How many study exactly one subject?

Question 10

A survey of 200 people found: 120 like coffee, 90 like tea, 85 like juice. 50 like both coffee and tea, 45 like both coffee and juice, 40 like both tea and juice. 25 like all three. How many like exactly two beverages?

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Exercise Set 5: Relation Properties

Question 11

Define relation R on set of integers as: $(a,b)\in R$ if $a\equiv b(\mathrm{mod}3)$. Which properties does R have?

Question 12

On the set of real numbers, define $(a,b)\in S$ if $a\le b$. Which properties does S have?

Question 13

For set $A=\{1,2,3,4\}$, relation $T=\{(1,1),(2,2),(3,3),(4,4),(1,2),(2,1)\}$. Which properties does T have?

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Exercise Set 6: Function Classification

Question 14

Function $f:\mathbb{Z}\to \mathbb{Z}$ defined by $f(x)=2x+1$. Classify as injective, surjective, or bijective.

Question 15

Function $g:\mathbb{R}\to {\mathbb{R}}^{+}$ defined by $g(x)=x^2$. Classify as injective, surjective, or bijective.

Question 16

Function $h:\{1,2,3,4\}\to \{a,b,c,d\}$ given by $h(1)=a,h(2)=b,h(3)=c,h(4)=d$. Classify.

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Exercise Set 7: Complex Set Operations

Question 17

Given:

• $A=\{x\in \mathbb{N}\mid x\mathrm{mod}3=0,1\le x\le 15\}$
• $B=\{x\in \mathbb{N}\mid x\mathrm{mod}5=0,5\le x\le 20\}$
• $C=\{x\in \mathbb{N}\mid x\mathrm{mod}7=0,7\le x\le 21\}$

Find $|(A\setminus (B\cup C))\cup (B\setminus (C\cup A))\cup (C\setminus (A\cup B))|$.

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Exercise Set 8: Real-world Relations

Question 18

In a family: Raj has 3 sons (Amit, Bob, Chris). Amit has 2 sons (David, Evan). Bob has 1 son (Frank). Let $M$ be all family members. Define:

• $R=\{(A,B)\mid A\text{ and }B\text{ are cousins}\}$
• $S=\{(A,B)\mid A\text{ is son of }B\}$

If $m=|R|$ and $n=|S|$, find $m+n$.

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💡 DETAILED SOLUTIONS

Solution to Question 1

Approach: Simplify each expression by rationalizing or expanding

1. $(\sqrt{12}+\sqrt{3})(\sqrt{8}-\sqrt{2})$ $=(2\sqrt{3}+\sqrt{3})(2\sqrt{2}-\sqrt{2})$ $=3\sqrt{3}\times \sqrt{2}=3\sqrt{6}$ (irrational)

2. $(\sqrt{18}-\sqrt{8})(\sqrt{6}+\sqrt{2})$ $=(3\sqrt{2}-2\sqrt{2})(\sqrt{6}+\sqrt{2})$ $=\sqrt{2}(\sqrt{6}+\sqrt{2})=\sqrt{12}+2=2\sqrt{3}+2$ (irrational)

3. $\frac{\sqrt{10}+\sqrt{5}}{\sqrt{10}-\sqrt{5}}$ Multiply numerator and denominator by $(\sqrt{10}+\sqrt{5})$: $=\frac{(\sqrt{10}+\sqrt{5}{)}^2}{10-5}=\frac{10+5+2\sqrt{50}}{5}=3+2\sqrt{2}$ (irrational)

4. $\frac{\sqrt{12}+\sqrt{3}}{\sqrt{12}-\sqrt{3}}$ $=\frac{2\sqrt{3}+\sqrt{3}}{2\sqrt{3}-\sqrt{3}}=\frac{3\sqrt{3}}{\sqrt{3}}=3$ (rational)

Answer: Options 1, 2, and 3 are irrational.

Solution to Question 4

Function: $f(x)=\frac{\sqrt{x^2-25}}{x-5}$

Domain restrictions:

1. $x^2-25\ge 0\Rightarrow x\le -5$ or $x\ge 5$
2. $x-5\ne 0\Rightarrow x\ne 5$

Integer domain: $\{...,-7,-6,-5,6,7,8,...\}$

Excluded integers: All integers from -4 to 4, plus 5 $A=\{-4,-3,-2,-1,0,1,2,3,4,5\}$

$|A|=10$

Solution to Question 7

Set: $S=\{1,2,3,...,24\}$

Relation R₁: $y=2x$ Valid pairs: $(1,2),(2,4),(3,6),...,(12,24)$ $|R_1|=12$

Relation R₂: $y=x^2$
Valid pairs: $(1,1),(2,4),(3,9),(4,16)$ $|R_2|=4$

Intersection: Pairs satisfying both conditions $(2,4)$ is the only pair where $y=2x=x^2$ $|R_1\cap R_2|=1$

Required: $|R_1\setminus (R_1\cap R_2)|=|R_1|-|R_1\cap R_2|=12-1=11$

Solution to Question 9

Given:

• $|M|=80$, $|P|=70$, $|C|=60$
• $|M\cap P|=40$, $|M\cap C|=35$, $|P\cap C|=30$
• $|M\cap P\cap C|=20$

Exactly one subject: $|M\text{ only}|=|M|-|M\cap P|-|M\cap C|+|M\cap P\cap C|$ $=80-40-35+20=25$

$|P\text{ only}|=|P|-|M\cap P|-|P\cap C|+|M\cap P\cap C|$ $=70-40-30+20=20$

$|C\text{ only}|=|C|-|M\cap C|-|P\cap C|+|M\cap P\cap C|$ $=60-35-30+20=15$

Total exactly one: $25+20+15=60$

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🖥️ COMPUTATIONAL THINKING

Algorithm for Checking Irrational Numbers

def is_irrational_expression(expr):
    """
    Check if mathematical expression results in irrational number
    """
    # Simplify the expression
    simplified = simplify_expression(expr)
 
    # Check if result can be written as p/q
    if is_rational(simplified):
        return False
    else:
        return True
 
def simplify_radical_product(expr):
    """
    Simplify expressions like (√a + √b)(√c - √d)
    """
    # Use distributive property
    # Apply √a × √b = √(ab)
    # Combine like terms
    pass

Set Operations Algorithm

def complex_set_operation(A, B, C):
    """
    Compute (A\(B∪C)) ∪ (B\(C∪A)) ∪ (C\(A∪B))
    """
    result1 = set_difference(A, set_union(B, C))
    result2 = set_difference(B, set_union(C, A))
    result3 = set_difference(C, set_union(A, B))
 
    return set_union(set_union(result1, result2), result3)

Function Domain Finder

def find_domain_function(f, domain_type='integers'):
    """
    Find domain of function with restrictions
    """
    domain = set()
 
    if domain_type == 'integers':
        for x in range(-100, 101):  # Reasonable range
            if is_valid_input(f, x):
                domain.add(x)
 
    return domain

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📊 VISUAL AIDS

Mermaid Diagram: Number System Hierarchy

graph TD
    A[Real Numbers ℝ] --> B[Rational Numbers ℚ]
    A --> C[Irrational Numbers]
    B --> D[Integers ℤ]
    B --> E[Non-integer Rationals]
    D --> F[Whole Numbers]
    D --> G[Negative Integers]
    F --> H[Natural Numbers ℕ]
    F --> I[Zero]

Mermaid Diagram: Function Types

graph LR
    A[Function] --> B[Injective?]
    B -->|Yes| C[Surjective?]
    B -->|No| D[Not Injective]
    C -->|Yes| E[Bijective]
    C -->|No| F[Injective Only]
    D --> G[Neither]

Mermaid Diagram: Relation Properties

graph TD
    A[Relation] --> B[Reflexive?]
    A --> C[Symmetric?]
    A --> D[Transitive?]
    B --> E[All Three = Equivalence]
    C --> E
    D --> E

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⚠️ COMMON PITFALLS AND TRAPS

Pitfall 1: Irrational Number Operations

Trap: Assuming $\sqrt{a}+\sqrt{b}$ is always irrational Reality: Sometimes $\sqrt{a}+\sqrt{b}$ can be rational Example: $\sqrt{4}+\sqrt{9}=2+3=5$ (rational)

Pitfall 2: Function Domain Restrictions

Trap: Forgetting that $\sqrt{x^2}=|x|$, not just $x$ Example: $\sqrt{x^2-16}$ requires $x^2-16\ge 0$, not $x-4\ge 0$

Pitfall 3: Set Operation Order

Trap: Not following proper order of operations Remember: Parentheses first, then set operations from left to right

Pitfall 4: Relation Properties

Trap: Confusing symmetric with antisymmetric Remember:

• Symmetric: $(a,b)\in R\Rightarrow (b,a)\in R$
• Antisymmetric: $(a,b)\in R$ and $(b,a)\in R\Rightarrow a=b$

Pitfall 5: Function Classification

Trap: Assuming all functions from ℝ to ℝ are bijective Reality: Most functions are neither injective nor surjective

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📋 QUICK REFRESHER HANDBOOK

Number Systems - Quick Facts

• Rational: Can be written as p/q, terminating or repeating decimals
• Irrational: Cannot be written as p/q, non-terminating non-repeating
• Operations: Product of irrationals can be rational

Set Theory - Formulas

• Inclusion-Exclusion: $|A\cup B|=|A|+|B|-|A\cap B|$
• Three Sets: $|A\cup B\cup C|=|A|+|B|+|C|-|A\cap B|-|A\cap C|-|B\cap C|+|A\cap B\cap C|$

Relations - Properties Check

1. Reflexive: Check $(a,a)$ for all $a$
2. Symmetric: If $(a,b)$ exists, check $(b,a)$
3. Transitive: If $(a,b)$ and $(b,c)$ exist, check $(a,c)$

Functions - Domain Rules

• Square root: Inside must be ≥ 0
• Denominator: Cannot be 0
• Logarithm: Inside must be > 0
• Combined functions: Intersection of individual domains

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🎯 PRACTICE TEST

Test Questions (Time: 45 minutes)

1. Which are irrational: $(\sqrt{6}+\sqrt{2})(\sqrt{3}-\sqrt{1})$, $\frac{\sqrt{8}+\sqrt{2}}{\sqrt{8}-\sqrt{2}}$?

2. Domain of $f(x)=\frac{\sqrt{x^2-49}}{x+7}$ for integers. Find excluded count.

3. For $S=\{n\in \mathbb{N}|n\le 30\}$, $R_1:y=2x$, $R_2:y=x^2$. Find $|R_1\setminus R_2|$.

4. Survey: 100 like cricket, 80 football, 70 tennis. 50 like both cricket and football, 40 both cricket and tennis, 30 both football and tennis, 20 all three. How many like exactly one?

5. Relation on integers: $(a,b)\in R$ if $a+b$ is even. Properties?

6. Function $f:\mathbb{Z}\to \mathbb{Z}$, $f(x)=x^2+1$. Classification?

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Answer Key

1. First expression irrational, second rational
2. 14 excluded integers
3. 13
4. 70
5. Reflexive, symmetric, transitive (equivalence)
6. Not injective, not surjective

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📈 Study Tips:

• Practice simplifying radical expressions
• Draw Venn diagrams for set problems
• Test relation properties systematically
• Remember domain restrictions for different function types
• Use real-world examples to understand abstract concepts