"Identifying infinite sets and determining cardinalities of finite sets"

Understanding Sets

  • Definition of a Set: A collection of distinct objects, considered as an object in its own right.

Finite vs Infinite Sets

  • A finite set has a countable number of elements. It can be empty or have a natural number cardinality.
  • An infinite set does not have a countable number of elements; it can be described with a pattern that does not end.

Cardinality

  • Cardinality: The number of elements in a set.
  • A set is finite if:
    • It is empty ($ ext{cardinality} = 0$).
    • Its cardinality is a natural number (1, 2, 3, …).
  • A set is infinite if:
    • The count of its elements never comes to a stop.

Examples of Finite Sets

  • Empty Set: $ ext{∅}$ - finite with cardinality 0.
  • Set of integers from 1 to 500: {1, 2, 3, …, 500} - finite with cardinality 500.
  • Set of specific numbers: {3, 7, 9, 11, 13, 17, 19} - finite with cardinality 7.

Examples of Infinite Sets

  • The set of all even numbers: {2, 4, 6, 8, …} - infinite.
  • The set of all integers greater than 15 and less than 20 is finite, but examples like {15, 16, 17, …} suggest counting does not stop - hence infinite in another definition.

Identifying Finite and Infinite Sets

  • If a count of elements eventually stops, it is a finite set.
  • If the count never stops, as indicated by patterns of ellipsis (….) or listing rules, it is an infinite set.

Practice Questions

  1. Determine whether the following sets are finite or infinite:
    • The set of integers between 1 and 100.
    • The set of natural numbers {1, 2, 3, …}.

Summary

  • Understand the definitions and the conditions that make a set finite or infinite.
  • Practice identifying sets through examples and different scenarios to solidify the concepts of cardinality and set types.