Cardinality - Examples and Properties

Examples and Properties

  • If X = {a, b, c} and Y = {apples, oranges, peaches}, then | X | = | Y | because {(a, apples), (b, oranges), (c, peaches)} is a bijection between the sets X and Y. The cardinality of each of X and Y is 3.
  • If | X | < | Y |, then there exists Z such that | X | = | Z | and Z ⊆ Y.
  • If | X | ≤ | Y | and | Y | ≤ | X |, then | X | = | Y |. This holds even for infinite cardinals, and is known as Cantor–Bernstein–Schroeder theorem.
  • Sets with cardinality of the continuum

Read more about this topic:  Cardinality

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