Lecture Notes: Define and Recognize the Empty Set

Introduction

• Objective: Learn to define and recognize the empty set.

Definition of the Empty Set

• Empty Set (Null Set):
  • A set that contains no elements.
  • Represented by two symbols:
    • Brackets: [ ] indicating nothing inside.
    • Empty Set Symbol: ∅, which looks like a zero with a slash through it.

Set Notation

• Important to distinguish between different notations:
  • Correct: [ ] or ∅ both represent an empty set.
  • Incorrect: {∅} represents a set containing one element, which is the empty set itself.

Recognizing the Empty Set

• Examples:
  • Example a: Set containing zero {0}.
    • Contains one element, zero.
    • Not an empty set.
  • Example b: 0 as a standalone.
    • Zero is a number, not a set.
    • Not an empty set but represents the number of members in an empty set.
  • Example c: The set of all ( x ) such that ( x < -3 ) or ( x > -5 ).
    • Contains numbers like -10 or 2.
    • Not an empty set because it contains elements.
  • Example d: The set of all ( x ) such that ( x ) is a rhombus with exactly five sides.
    • A rhombus is defined as a four-sided figure.
    • An empty set because a rhombus cannot have five sides.