Quick Proof

A request from ezgineer inspired me to to talk about the proof that there are more real numbers than natural numbers.

This statement was proved by Cantor, who showed that the set of real numbers was not countable and that there were more of them than natural numbers. He used the technique of proof by contradiction. His argument (called Cantors Diagonalisation Argument) goes as follows:

Suppose the real numbers are countable, you could set up a one-to-one correspondence between them (R) and the natural numbers (N). Then we could match up each natural number with a real number.

matching_4.gif

Next, we create a number that is not on the list:

matching_5.gif

We have created a new real number – o.683175…- that does not correspond to any natural number. Thus, this list in incomplete and this there is no one-to-one correspondence between N and R. We have therefore proved the statement by contradiction.

This idea of the different sizes of infinity was given the name of cardinality; we say that the cardinality of two sets A and B are the same – written card(A) = card(B) – if there is a one-to-one correspondence between these two sets.

Any other suggestions of proofs you would like me to write about? M x

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