Set Theory: Equality and Equivalence of Sets

Key Concepts

• Set Equality:

  • Two sets are considered equal if they contain exactly the same elements.
  • Important distinction: Only distinct elements are counted.
  • Sets may appear different due to the order or repetition of elements but can still be equal if they contain the same distinct elements.
  • Example:
    • Set A = {1, 2, 3}
    • Set B = {3, 2, 1}
    • Both sets are equal because they have the same elements.

• Set Equivalence:

  • Sets are considered equivalent if they have the same number of distinct elements, regardless of the actual elements themselves.
  • Repetition of elements does not affect equivalence, only the count of distinct elements.
  • Example:
    • Set C = {a, b, c}
    • Set D = {x, y, z}
    • Both sets are equivalent because they each have three distinct elements.

• Neither Equal Nor Equivalent:

  • If two sets do not have the same number of distinct elements, they are neither equal nor equivalent.

Summary

Understanding the difference between equal and equivalent sets is crucial in set theory. Equal sets require identical elements, whereas equivalent sets require the same number of distinct elements but not necessarily the same elements. Repetition and order do not impact these definitions. This knowledge helps in accurately determining the relationships between different sets.