Lecture Notes on Sets and Related Concepts

Introduction to Sets

• Discussion on the definition and examples of sets
• Classification of sets:
  • Well-defined sets
  • Empty sets (null sets)

Key Terminology

• Element: An individual object or member of a set.
• Cardinality: The number of elements in a set.
• Well-defined set: A set with clearly defined elements.
• Empty set: A set with no elements, denoted by {} or ∅.

Types of Numbers and Their Sets

1. Integers: Negative numbers, zero, and positive numbers.

   • Examples: -3, -2, -1, 0, 1, 2, 3

2. Whole Numbers: Non-negative integers including zero.

   • Examples: 0, 1, 2, 3, ...

3. Natural Numbers (Counting Numbers): Positive integers starting from 1.

   • Examples: 1, 2, 3, ...

4. Even Numbers: Integers divisible by 2.

   • Examples: 0, 2, 4, 6, ...

5. Odd Numbers: Integers not divisible by 2.

   • Examples: 1, 3, 5, 7, ...

6. Prime Numbers: Numbers greater than 1 with exactly two distinct positive divisors (1 and themselves).

   • Examples: 2, 3, 5, 7, 11, ...

7. Composite Numbers: Numbers with more than two divisors.

   • Examples: 4, 6, 8, 9, 10, ...

8. Perfect Squares: The product of an integer multiplied by itself.

   • Examples: 1, 4, 9, 16, ...

9. Multiples: Products of a number and integers.

   • Examples: Multiples of 3: 3, 6, 9, ...
   • Examples: Multiples of 5: 5, 10, 15, ...

10. Factors: Numbers that divide another number without leaving a remainder.

    • Example: Factors of 6: 1, 2, 3, 6

Example Groupings

• Grouping by Characteristics:
  • Group A: Shoes, jacket, cup (items that can be worn).
  • Group B: Ball, doll, toy car (toys).
  • Group C: Orange, mango, banana (fruits).

Well-Defined vs. Not Well-Defined Sets

• Well-defined set examples:
  • Set of primary colors: {red, blue, yellow}
  • Set of even numbers: {2, 4, 6, ...}
• Not well-defined set examples:
  • Set of beautiful girls in school (subjective)
  • Set of popular singers (subjective)

Cardinality

• Denoted by "n" followed by the set name in parentheses (e.g., n(A)).
• Example: For set A (primary colors: {red, blue, yellow}), cardinality n(A) = 3.
• Empty set has cardinality n(∅) = 0.

Examples of Sets and Their Cardinalities

1. Set A: Days of the week in school (Monday to Friday).
   • Elements: {Monday, Tuesday, Wednesday, Thursday, Friday}
   • Cardinality: n(A) = 5
2. Set B: Counting numbers less than 5.
   • Elements: {1, 2, 3, 4}
   • Cardinality: n(B) = 4
3. Set C: Set of days in a week starting with 'M'.
   • Element: {Monday}
   • Cardinality: n(C) = 1

Conclusion

• Understanding sets is crucial for math and science.
• Importance of recognizing well-defined and non-well-defined sets.