Introduction to Sets

Definition of Sets

• A set is an unordered collection of objects.
  • Example: The set of brothers includes Eric, Adam, Kevin (order does not matter).
• An element is an object within a set.
  • Example: Elements of set B are Eric, Adam, Kevin.

Notation

• Element of a Set
  • Notation: ( \in ) means "is an element of".
  • Example: Adam ( \in ) B (Adam belongs to set B).
  • Notation: ( \notin ) means "is not an element of".
  • Example: Larry ( \notin ) B (Larry does not belong to set B).

Important Sets

1. Natural Numbers
   • Counting numbers: 1, 2, 3, ...
2. Whole Numbers
   • Natural numbers including 0: 0, 1, 2, 3, ...
3. Integers
   • Positive and negative whole numbers: ..., -2, -1, 0, 1, 2, ...
4. Rational Numbers (Q)
   • Quotients of integers: ( \frac{a}{b} ), where ( b \neq 0 ).
5. Real Numbers (R)
   • Include all natural, whole, integers, and rational numbers.
6. Complex Numbers (C)
   • Numbers in form ( a + bi ), where ( a ) is real, ( bi ) is imaginary.
7. Special Integers Notation
   • ( \mathbb{Z}^{+} ) or ( \mathbb{Z}^{-} ) for positive/negative integers.

Set Notation

• Roster Notation
  • Listing elements directly: ( S = {1, 2, 3, 4, 5} ).
  • Used for discrete sets (countable objects).
  • Not suitable for continuous sets (e.g., values between 0 and 1).
• Set Builder Notation
  • Describes elements with conditions: ( S = { x | x \in \mathbb{N}, x \leq 5 } ).
  • Suitable for both discrete and continuous sets.
• Interval Notation
  • Used for continuous sets.
  • Example for set B: ( [0, 1] ) (includes 0 and 1).
  • Uses brackets (([])) for inclusive, parentheses ((())) for exclusive.

Special Sets

• Universal Set
  • Set of all elements under consideration.
  • Example: Venn diagram with natural numbers as universal set.
• Empty Set
  • Set with no elements, denoted by ( \emptyset ) or ( { } ).
  • Note: ( { \emptyset } ) is not the empty set but a set containing the empty set.

Next Topics

• Exploration of set relationships and subsets.