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Understanding the Concept of Empty Set

Oct 26, 2024

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.