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add section on infinite sets
This suggests another definition for cardinality, that two sets have the same cardinality if there is a one-to-one correspondence between them. This is the definition we will use for infinite sets, where the concept of the number of elements of a set is not as clear.
This suggests another definition for cardinality, that two sets have the same cardinality if there is a one-to-one correspondence between them. This is the definition we will use for infinite sets, where the concept of the number of elements of a set is not as clear.
==Infinite sets==
Infinite sets are sets that go on for ever. For example, the set of natural numbers (1,2,3...) is infinite, as no matter how far we go, there will always be more numbers. An important type of infinite set is the countable set. A set is [[countable]], if it can be written as a list. For example, the set of natural numbers itself is countable. This is where countable sets get their name from, as the natural numbers are sometimes called counting numbers, as they are the numbers we use to count with.
One alternative definition of a set being countable, is that it has the same cardinality as the natural numbers. This is equivalent to saying that there is a one-to-one correspondence between the set, and the set of natural numbers.
One example of a countable set is the set of even numbers. These can be written as a list: 2, 4, 6, 8, 10, 12, 14, 16... Alternatively, they can be put into a one-to-one correspondence with the set of natural numbers: 1 <-> 2, 2 <-> 4, 3 <-> 6, 4 <-> 8, 5 <-> 10,....
Similarly, the multiples of 3 (3, 6, 9, 12, 15...), 4 (4, 8, 12, 16, 20, 24, 28... and so on, each form a countable set, as each can be written as a list.
One unusual property of infinite sets is that a "smaller" set can have the same cardinality as a "larger" one. For example here we see that the set of even numbers has the same cardinality as the set of natural numbers, and thus it is in a sense the same size, even though the set of even numbers misses out a lot of numbers from the set of natural numbers.
[[category: set theory]]
[[category: set theory]]