Set Theory and Number Systems

1. Core Concepts & Intuition

(Imagine you have a box of crayons. That box is a set, and each crayon is an element of that set. Set theory is just a way to formally talk about grouping things together. We can have a set of numbers, a set of letters, or even a set of other sets! It’s the fundamental way we organize and categorize information in mathematics. Number systems are just specific, very important sets of numbers that we use for counting and measuring.)

2. Formal Definitions, Jargon, and Nuances

A set is a well-defined collection of distinct objects. The objects that make up the set are called its elements or members.

A subset (written as A ⊆ B) is a set where all the elements of set A are also elements of set B.

1. Natural Numbers (ℕ): {1, 2, 3, 4, …} - The counting numbers.
2. Whole Numbers (W): {0, 1, 2, 3, …} - Natural numbers plus zero.
3. Integers (ℤ): {…, -3, -2, -1, 0, 1, 2, 3, …} - Whole numbers and their opposites.
4. Rational Numbers (ℚ): Numbers that can be expressed as a fraction p/q of two integers, where q ≠ 0. (e.g., 1/2, -3/4, 7).
5. Irrational Numbers: Numbers that cannot be expressed as a simple fraction (e.g., π, √2).
6. Real Numbers (ℝ): All rational and irrational numbers.

Set Operations:

• Union (A ∪ B): The set of all elements that are in A, or in B, or in both.
• Intersection (A ∩ B): The set of all elements that are in both A and B.
• Difference (A - B): The set of all elements that are in A but not in B.
• Complement (A’): The set of all elements not in A (relative to a universal set U).

3. Step-by-Step Procedures & Worked Examples

Procedure: Finding the Union and Intersection of Two Sets

1. List the elements of both sets.
2. For Union (∪): Combine all unique elements from both sets into a new set.
3. For Intersection (∩): Identify only the elements that appear in both sets.

Example 1: Let A = {1, 2, 3, 4} and B = {3, 4, 5, 6}.

• A ∪ B: {1, 2, 3, 4, 5, 6}
• A ∩ B: {3, 4}

Example 2: Let X = {a, b, c} and Y = {x, y, z}.

• X ∪ Y: {a, b, c, x, y, z}
• X ∩ Y: {} (The empty set, as there are no common elements)

Visual Representation (Venn Diagram)

graph TD
    subgraph Universal Set
        subgraph Set A
            1
            2
        end
        subgraph Set B
            5
            6
        end
        subgraph Intersection
            3
            4
        end
    end

4. The “Exam Brain” Algorithm & Strategic Handbook

Pattern Recognition:

• Keywords to look for: “collection of”, “distinct elements”, “belongs to”, “union”, “intersection”, “subset”, “what type of number”.
• Question Formats: “Which of the following is a rational number?”, “Find the union/intersection of sets A and B”, “If A is a subset of B, what can we conclude?“.

Mental Algorithm (The Approach):

1. Identify the Goal: Am I being asked to classify a number, or perform an operation on sets?
2. Select the Tool:
   • For number classification, use the definitions (ℕ, ℤ, ℚ, ℝ).
   • For set problems, identify the operation required (∪, ∩, -, etc.).
3. Execute & Verify:
   • For fractions, check if the denominator is zero.
   • For set operations, go element by element to avoid missing any. Use a Venn diagram mentally or on paper to visualize.

5. Common Pitfalls & Exam Traps

• Trap 1: Confusing Subsets and Proper Subsets. A set is always a subset of itself. A proper subset (⊂) means it’s a subset but not equal to the original set.
• Trap 2: Forgetting Zero. Forgetting that 0 is a whole number but not a natural number, and that it is an even integer.

6. Practice Exercises (Scaffolded Difficulty)

Exercise 1 (Concept Check)

True or False: Every integer is a rational number.

Exercise 2 (Application)

Given Set A = {x | x is a prime number less than 10} and Set B = {y | y is an even number less than 10}, find A ∩ B.

Exercise 3 (Qualifier-Style Synthesis)

Let S be the set of all rational numbers between 0 and 1. Which of the following statements is true? (a) S is a finite set. (b) There is a “largest” number in S that is less than 1. (c) S contains a finite number of elements. (d) S is an infinite set.

7. Comprehensive Solutions to Exercises

Solution 1:

True. Any integer ‘z’ can be written as the fraction z/1, which fits the definition of a rational number.

Solution 2:

1. Identify the Goal: Find the intersection of two sets.
2. Select the Tool: The intersection (∩) contains elements common to both sets.
3. Execute & Verify:
   • First, list the elements:
     • Set A (prime numbers < 10): {2, 3, 5, 7}
     • Set B (even numbers < 10): {2, 4, 6, 8}
   • The only common element is 2.
   • Therefore, A ∩ B = {2}.

Solution 3:

(d) S is an infinite set. Between any two distinct rational numbers, you can always find another rational number (e.g., by taking their average). For example, between 1/2 and 1/3, there is (1/2 + 1/3)/2 = 5/12. Because this process can be repeated infinitely, there are infinite rational numbers between 0 and 1.

• (a) and (c) are false because the set is infinite.
• (b) is false because for any rational number r < 1, you can always find another rational number (r+1)/2 which is also less than 1 but greater than r. So there is no “largest” number.

---

8. Connections & Further Learning

• [[Relations and Functions]]
• [[Number Theory]]
• [[Logic]]

---