🧮 Mathematics 1 - Week 1: Set Theory & Relations

“Mathematics is the language of patterns. Master the patterns, master the subject.”

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📋 Week 1 Overview

Week 1 focuses on foundational concepts that appear throughout the course:

• Number Systems - Understanding rational vs irrational numbers
• Functions - Domain, range, and properties
• Set Theory - Operations, cardinality, Venn diagrams
• Relations - Properties and equivalence relations

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🔢 Pattern 1: Number Systems & Irrational Numbers

📖 Concept Explanation

Rational Numbers: Can be expressed as $\frac{p}{q}$ where $p,q\in \mathbb{Z},q\ne 0$ Irrational Numbers: Cannot be expressed as a simple fraction

🧠 Mental Algorithm

1. Simplify the expression completely
2. Check if it can be written as $\frac{p}{q}$
3. Look for square roots that don’t simplify to rational numbers

📝 Pattern-Based Examples

Example 1: Product of Irrational Expressions

Question Pattern: $(\sqrt{a}\pm \sqrt{b})(\sqrt{c}\mp \sqrt{d})$

Problem: Determine if $(\sqrt{8}+\sqrt{2})(\sqrt{12}-\sqrt{3})$ is rational or irrational.

Solution:

$$(\sqrt{8}+\sqrt{2})(\sqrt{12}-\sqrt{3})=\sqrt{8}\sqrt{12}+\sqrt{2}\sqrt{12}-\sqrt{8}\sqrt{3}-\sqrt{2}\sqrt{3}$$

Step-by-step:

1. $\sqrt{8}=2\sqrt{2}$, $\sqrt{12}=2\sqrt{3}$
2. $\sqrt{2}\sqrt{12}=\sqrt{2}\cdot 2\sqrt{3}=2\sqrt{6}$
3. $\sqrt{8}\sqrt{3}=2\sqrt{2}\cdot \sqrt{3}=2\sqrt{6}$
4. Result: $2\sqrt{6}$ (irrational)

Example 2: Rationalizing Complex Fractions

Question Pattern: $\frac{\sqrt{a}\pm \sqrt{b}}{\sqrt{a}\mp \sqrt{b}}$

Problem: Simplify $\frac{\sqrt{6}+\sqrt{3}}{\sqrt{6}-\sqrt{3}}$

Solution:

$$\frac{\sqrt{6}+\sqrt{3}}{\sqrt{6}-\sqrt{3}}\cdot \frac{\sqrt{6}+\sqrt{3}}{\sqrt{6}+\sqrt{3}}=\frac{(\sqrt{6}+\sqrt{3}{)}^2}{6-3}=\frac{6+2\sqrt{18}+3}{3}=\frac{9+6\sqrt{2}}{3}=3+2\sqrt{2}$$

Answer: $3+2\sqrt{2}$ (irrational)

⚠️ Common Pitfalls

• Forgetting to simplify square roots first
• Missing the cross terms when multiplying conjugates
• Assuming $\sqrt{a}+\sqrt{b}$ is always irrational (it’s rational when $a=b$)

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🔄 Pattern 2: Domain of Functions

📖 Concept Explanation

Domain: Set of all possible input values (x) for which f(x) is defined Common Restrictions:

• Division by zero: $x\ne$ values making denominator zero
• Square roots: Expression under radical ≥ 0
• Logarithms: Argument > 0

🧠 Mental Algorithm

1. Identify all restrictions in the function
2. Solve each restriction inequality
3. Find intersection of valid regions
4. Count excluded values if domain ⊂ ℤ

📝 Pattern-Based Examples

Example 3: Square Root Functions

Question Pattern: $f(x)=\frac{\sqrt{x^2-a}}{x\pm b}$ where domain ⊂ ℤ

Problem: Find domain of $f(x)=\frac{\sqrt{x^2-16}}{x+4}$ where $D\subset \mathbb{Z}$

Solution:

$$\sqrt{x^2-16}\ge 0⟹x^2-16\ge 0⟹|x|\ge 4⟹x\le -4\text{ or }x\ge 4x+4\ne 0⟹x\ne -4$$

Valid domain: $x\le -4$ (since x ≥ 4 and x ≠ -4 excludes x = 4)

Integers: x = -4 is excluded, so x ≤ -5 Set A (excluded integers): x ≥ 5 (since |x| ≥ 4 but x ≠ 4) Cardinality: Infinite (not a finite set)

Wait - Domain ⊂ ℤ means we need to find integers NOT in domain:

For $f(x)=\frac{\sqrt{x^2-16}}{x+4}$, when x = -4:

• Numerator: $\sqrt{(-4{)}^2-16}=\sqrt{0}=0$
• Denominator: -4 + 4 = 0
• Division by zero → undefined

Valid integers: x ≤ -5 or x ≥ 5 Set A (integers NOT in domain): x = -4 only Cardinality: 1

But wait, let me double-check the actual assignment answer…

Actually, looking at the assignment: “Let A be the set of integers which are not in the domain of f”

The function is undefined when:

1. $x^2-16<0$ (inside square root negative)
2. $x+4=0$ (division by zero)

For domain ⊂ ℤ:

• Square root domain: x ≤ -4 or x ≥ 4
• But x ≠ -4 (division by zero)

So integers in domain: x ≤ -5 or x ≥ 5 Integers NOT in domain: x = -4, -3, -2, -1, 0, 1, 2, 3, 4

That’s 9 values, but the answer was 8. Let me think…

Actually, when x = 4:

• Numerator: $\sqrt{16-16}=0$
• Denominator: 4 + 4 = 8
• Function value: 0/8 = 0 (defined!)

So x = 4 IS in the domain. The division by zero only happens at x = -4.

Correct Set A: x = -4, -3, -2, -1, 0, 1, 2, 3 (8 values) Answer: 8

Example 4: Function Domain Intersection

Question Pattern: Domain of sum/difference of functions

Problem: If f₁(x) has domain D₁ and f₂(x) has domain D₂, find domain of f₁(x) + f₂(x)

Standard Result: $D_1\cap D_2$

Proof: For (f₁ + f₂)(x) = f₁(x) + f₂(x) to be defined:

• f₁(x) must be defined → x ∈ D₁
• f₂(x) must be defined → x ∈ D₂
• Therefore x ∈ D₁ ∩ D₂

⚠️ Common Pitfalls

• Forgetting division by zero restrictions
• Not considering domain ⊂ ℤ constraint
• Confusing |x| ≥ a vs |x| > a for integer solutions

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📊 Pattern 3: Set Operations & Cardinality

📖 Concept Explanation

Set Operations:

• Union (∪): A ∪ B = {x | x ∈ A or x ∈ B}
• Intersection (∩): A ∩ B = {x | x ∈ A and x ∈ B}
• Difference (A\B): A\B = {x | x ∈ A and x ≠ B}
• Complement: A’ = {x | x ∉ A}

Cardinality: |A| = number of elements in set A

🧠 Mental Algorithm

1. Draw Venn Diagram mentally or on paper
2. Apply inclusion-exclusion for multiple sets
3. Use systematic counting for complex conditions
4. Verify with small cases

📝 Pattern-Based Examples

Example 5: Complex Set Operations

Question Pattern: |AΔ(B∪C)| or |(A∖B)∪(B∖C)∪(C∖A)|

Problem: Find |(A∖(B∪C)) ∪ (B∖(C∪A)) ∪ (C∖(B∪A))|

Solution Strategy: This is the symmetric difference pattern:

• Elements in exactly one of A, B, C
• Formula: |AΔBΔC| = |A∪B∪C| - |A∩B∩C| - |A∩B∖C| - |A∩C∖B| - |B∩C∖A|

For finite sets with conditions:

1. Count elements satisfying each individual condition
2. Use Venn diagram regions
3. Apply inclusion-exclusion principle

Example 6: Venn Diagram Applications

Question Pattern: Survey problems with multiple categories

Problem: In a zoo with 6 white tigers and 6 royal tigers, 5 are males, 10 are either royal or male. Find female white tigers.

Solution:

Let W = white tigers, R = royal tigers, M = male tigers Given: |W| = 6, |R| = 6, |M| = 5, |R∪M| = 10 Find: |W∖M| = |W| - |W∩M| Using Venn diagram: - |R∪M| = |R| + |M| - |R∩M| = 6 + 5 - |R∩M| = 10 - 11 - |R∩M| = 10 ⇒ |R∩M| = 1 Total tigers: 12 - |W∪M| = |W| + |M| - |W∩M| = 6 + 5 - |W∩M| = 12 - |W∖M| - |W∪M| = |W∖M| + |W∩M| = |W∖M| + 1 But |W∪M| = total - |R∖M| = 12 - 5 = 7 - 7 = |W∖M| + 1 ⇒ |W∖M| = 6 Wait, this doesn't make sense. Let me recalculate... Actually, |W∪M| should include all white tigers and all males. But we have |R∪M| = 10, which includes royal tigers and males. The issue is I need to find the intersection properly. **Correct approach:** Total = |W∪R| = 12 |R∪M| = 10 |W∪M| = ? Using inclusion-exclusion for three sets: |W∪R∪M| = |W| + |R| + |M| - |W∩R| - |W∩M| - |R∩M| + |W∩R∩M| But we don't know |W∪R∪M|. **Alternative approach:** Let f = female white tigers (what we want) Then: - White females: f - White males: 6 - f - Royal females: total females - f - Royal males: total males - (6 - f) = 5 - (6 - f) = f - 1 Total females = f + (royal females) But |R∪M| = royal tigers + males = 6 + 5 - |R∩M| = 10 This is getting complicated. Let me use the standard method: **Standard Venn Diagram Method:** Let x = |W∩R∩M| Then: - |W∩R| = 6 - x (since |W| = 6, so non-male white tigers = 6 - x) - |W∩M| = 5 - x (since |M| = 5, so non-royal males = 5 - x) - |R∩M| = y (let’s call this y) Given |R∪M| = 10 = |R| + |M| - |R∩M| = 6 + 5 - y = 11 - y ⇒ y = 1 So |R∩M| = 1 Now, |W∖(R∪M)| = female white tigers = |W| - |W∩M| - |W∩R| + |W∩R∩M| |W∩M| = 5 - x |W∩R| = 6 - x |W∩R∩M| = x So |W∖(R∪M)| = 6 - (5 - x) - (6 - x) + x = 6 - 5 + x - 6 + x + x = x But this gives x, not a number. I need another equation. Total tigers = 12 = |W∪R∪M| = |W| + |R| + |M| - |W∩R| - |W∩M| - |R∩M| + |W∩R∩M| = 6 + 6 + 5 - (6 - x) - (5 - x) - 1 + x = 17 - 6 - 5 - 1 + x + x = 5 + 2x 12 = 5 + 2x ⇒ 2x = 7 ⇒ x = 3.5 This doesn't make sense for cardinalities. I think I made an error in the interpretation. Let me re-read the problem. "In a Zoo, there are 6 Bengal white tigers and 6 Bengal royal tigers. Out of these tigers, 5 are males and 10 are either Bengal royal tigers or males." The phrasing "10 are either Bengal royal tigers or males" means |R∪M| = 10 And there are 6 white + 6 royal = 12 total tigers. Now, 5 are males, so 12 - 5 = 7 are females. Let w = white females r = royal females Then: - White tigers: w + (5 - r) = 6 (since males = 5, so white males = 5 - r) - Royal tigers: r + (5 - w) = 6 (since royal males = 5 - w) |R∪M| = (white males + royal males) + (white females + royal females) - (males who are both white and royal) = (5 - r + 5 - w) + (w + r) - (males who are white and royal) Males who are white and royal = 5 - r (since white males = 5 - r, and if they are royal too, then royal males = 5 - w, but this is getting complicated. **Simpler approach using the given information directly:** From |R∪M| = 10 and |M| = 5, we have |R∩M| = |R| + |M| - |R∪M| = 6 + 5 - 10 = 1 So 1 tiger is both royal and male. Now, total males = 5, so 5 - 1 = 4 males are white but not royal. Total white tigers = 6, so white females = 6 - 4 = 2 **Answer: 2** ### ⚠️ Common Pitfalls - Misinterpreting "either A or B" vs "both A and B" - Forgetting inclusion-exclusion principle for three sets - Not accounting for empty intersections --- ## 🔗 Pattern 4: Relations and Their Properties ### 📖 Concept Explanation **Relation:** A set of ordered pairs (x, y) with some connection between x and y **Properties:** - **Reflexive:** (a, a) ∈ R for all a ∈ S - **Symmetric:** (a, b) ∈ R ⇒ (b, a) ∈ R - **Transitive:** (a, b) ∈ R and (b, c) ∈ R ⇒ (a, c) ∈ R - **Equivalence Relation:** Reflexive, symmetric, AND transitive ### 🧠 Mental Algorithm 1. **Test each property** systematically 2. **Find counterexamples** for properties that fail 3. **Check domain constraints** (especially for inequalities) ### 📝 Pattern-Based Examples #### Example 7: Numerical Relations **Question Pattern:** R = {(x,y) | |x-y| ≤ k} or 5 divides (x-y) **Problem:** R₁ = {(x,y): x,y ∈ ℤ, |x-y| ≤ 5} vs R₂ = {(x,y): x,y ∈ ℤ, 5 divides x-y} **Analysis:** - **R₂ is equivalence relation:** - Reflexive: |x-x| = 0, 5 divides 0 ✓ - Symmetric: 5 divides (x-y) ⇒ 5 divides (y-x) ✓ - Transitive: 5 divides (x-y) and 5 divides (y-z) ⇒ 5 divides (x-z) ✓ - **R₁ is not equivalence relation:** - Reflexive: |x-x| = 0 ≤ 5 ✓ - Symmetric: |x-y| ≤ 5 ⇒ |y-x| ≤ 5 ✓ - Transitive: |x-y| ≤ 5 and |y-z| ≤ 5, but |x-z| could be > 5 ✗ #### Example 8: Set-Based Relations **Question Pattern:** Relations on finite sets with cardinality constraints **Problem:** Survey data with multiple categories, relation defined by shared properties. **Solution Strategy:** 1. **Model as sets** A, B, C with given cardinalities 2. **Find intersections** using inclusion-exclusion 3. **Test each property** with specific counterexamples ### ⚠️ Common Pitfalls - Assuming |x-y| ≤ k implies transitivity - Forgetting domain constraints affect transitivity - Not testing with specific counterexamples --- ## 🏗️ Pattern 5: Functions (Injective, Surjective, Bijective) ### 📖 Concept Explanation **Injective (One-to-One):** f(a) = f(b) ⇒ a = b **Surjective (Onto):** For every y ∈ codomain, ∃ x ∈ domain with f(x) = y **Bijective:** Both injective and surjective ### 🧠 Mental Algorithm 1. **Check injective:** See if different inputs give different outputs 2. **Check surjective:** See if every possible output is achieved 3. **For finite sets:** Compare |domain| and |codomain| ### 📝 Pattern-Based Examples #### Example 9: Finite Set Functions **Question Pattern:** Function from materials to dielectric constants **Problem:** Materials → {1,2,3,7,8,13}, determine if bijective **Solution:** - **Check surjective:** Are all values {1,2,3,7,8,13} used? - **Check injective:** Does each material map to unique value? **Answer:** Bijective if one-to-one correspondence exists #### Example 10: Rational to Integer Functions **Question Pattern:** f: ℚ → ℤ with specific formulas **Problem:** f(p/q) = p - q where gcd(p,q) = 1 **Analysis:** - **Surjective:** Every integer can be written as p - q? - For target 1: 2-1, 3-2, 4-3, etc. ✓ - For target 0: 1-1, 2-2, etc. ✓ - For negative: 1-2 = -1 ✓ - **Injective:** f(a) = f(b) ⇒ a = b? **Answer:** Surjective but not injective (multiple representations for same value) ### ⚠️ Common Pitfalls - Confusing injective vs surjective for finite vs infinite sets - Not considering domain/codomain carefully - Assuming linear functions are always bijective --- ## 📚 Pattern-Based Exercises ### Set 1: Number Systems (4 questions) 1. $(\sqrt{10}+\sqrt{2})(\sqrt{5}-\sqrt{2})$ = ? 2. $\dfrac{\sqrt{12}+\sqrt{3}}{\sqrt{12}-\sqrt{3}}$ = ? 3. $(\sqrt{6}-\sqrt{2})^2$ = ? 4. $\dfrac{\sqrt{8}-\sqrt{2}}{\sqrt{8}+\sqrt{2}}$ = ? ### Set 2: Function Domains (3 questions) 1. Find domain of $f(x) = \sqrt{x^2 - 9}$ for x ∈ ℤ 2. Find domain of $g(x) = \dfrac{1}{\sqrt{x^2 - 4x}}$ for x ∈ ℤ 3. Find integers not in domain of $h(x) = \dfrac{\sqrt{x^2 - 25}}{x - 5}$ ### Set 3: Set Cardinality (4 questions) 1. |A∪B∪C| where |A|=10, |B|=8, |C|=6, |A∩B|=3, |A∩C|=2, |B∩C|=4, |A∩B∩C|=1 2. |(A∖B)∪(B∖A)| given |A|=15, |B|=12, |A∩B|=5 3. Number of elements in (A∩B')∪(A'∩B) given above 4. |AΔBΔC| for three sets with given cardinalities ### Set 4: Relations (3 questions) 1. Is R = {(x,y) | x² + y² = 25} reflexive on ℝ? 2. Is R = {(x,y) | x divides y} symmetric on ℕ? 3. Is R = {(x,y) | |x-y| ≤ 3} transitive on ℤ? ### Set 5: Functions Properties (3 questions) 1. f: {1,2,3} → {a,b,c} with f(1)=a, f(2)=b, f(3)=a - injective? 2. f: ℝ → ℝ with f(x) = x² - surjective? 3. f: ℤ → ℤ with f(x) = 2x - bijective? --- ## 🎯 Mental Algorithms Summary ### Quick Reference for Exam 1. **Irrational Numbers:** Simplify completely, rationalize denominators 2. **Domain:** Solve all restrictions, find intersection, count integers 3. **Sets:** Draw Venn diagram, use inclusion-exclusion, systematic counting 4. **Relations:** Test each property with counterexamples 5. **Functions:** Compare domain/codomain sizes, test injectivity/surjectivity --- ## 📈 Progress Tracking **Week 1 Completion Checklist:** - [ ] Master all 5 question patterns - [ ] Complete 15+ practice exercises - [ ] Achieve 90%+ accuracy on pattern sets - [ ] Review all solution explanations - [ ] Identify personal error patterns --- > **Next:** [Week 2 - Coordinate Geometry](./Week2-CoordinateGeometry.md) > > **Remember:** Patterns are your friends. Master them, and the questions become predictable.