#MathsInMinutes Day 24: Cardinality and countability

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The cardinality of a finite set A, written |A|, is the number of distinct elements within it. Two sets, whether finite of infinite, are said to have the same cardinality if their elements can be put into one-to-one correspondence. This means that elements of each set can be paired off, with each element associated with precisely one element on the other set.

Countable sets are those sets whose elements can be labelled by the natural numbers. Intuitively, this means that the set´s elements can be listed, although the list may be finite. Mathematically, it means the set can be put into one-to-one correspondence with a subset of the natural numbers.

This has surprising consequences. For instance, a strict subset of a countable set can have the same cardinality as the set itself. So, the set of all even numbers has the same cardinality as the set of square numbers, which has the same cardinality as the natural numbers. All are said to be countably infinite.

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