Finite vs Infinite Math Example 4

Follow the full solution, then compare it with the other examples linked below.

Example 4

medium
A hotel has infinitely many rooms, all occupied. A new guest arrives. Explain (Hilbert's Hotel) how the hotel can accommodate the guest without anyone leaving.

Solution

  1. 1
    Ask every current guest in room nn to move to room n+1n+1.
  2. 2
    Room 11 is now vacant, and every existing guest still has a room.
  3. 3
    The new guest takes room 11. All guests are accommodated.

Answer

Move guest in room nn to room n+1n+1 for all nn; the new guest takes room 11.
Hilbert's Hotel illustrates a key property of infinite sets: a countably infinite set can absorb finitely many new elements by 'shifting'. This works because there is no 'last room' to run out of — the bijection n↦n+1n \mapsto n+1 maps N\mathbb{N} to N∖{1}\mathbb{N} \setminus \{1\}, showing these subsets have the same cardinality.

About Finite vs Infinite

Finite describes a quantity or set with a definite end; infinite describes something that goes on forever without bound.

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More Finite vs Infinite Examples

Example 1 easy

Classify each set as finite or infinite and explain: (a) the days of the week, (b) the multiples of

Example 2 medium

Is the set of decimal numbers between [formula] and [formula] finite or infinite? Is it countable or

Example 3 easy

For each, say whether the process terminates (finite) or goes on forever (infinite): (a) counting to

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