The diagonalization principle

http://faculty.simpson.edu/lydia.sinapova/www/cmsc365/LN365_Lewis/L03-Countability.htm#diag

The diagonalization principle is based on a simple observation.

Let A be a finite set, and R be a binary relation on A. We can represent the relation by a square table, rows and columns representing the elements, and cells having 1s if there is a link between the corresponding elements.

For example, if A = {a,b,c,d}, and R = {(a,b), (a,c), (b,b), (b,d), (c,b), (c,d), (d,a)}, the table would be:

		a	b	c	d

	a	0	1	1	0

	b	0	1	0	1

	c	0	1	0	1

	d	1	0	0	0

The principle says that the compliment of the diagonal
(replacing 1s with 0s and vice versa) is different from each row.

The reversed diagonal in the example is 1,0,1,1 and you can see that it is different from each row.

Explanation: The reversed diagonal differs from the first row in the first element (we have taken the complement in the diagonal), it differs from the second row in the second element, etc.

Application of the diagonalization principle to prove uncountability of sets.
Basic idea: In order to apply the principle the elements of the set have to be represented as infinite sequences of 0 and 1, and any infinite sequence of 0 and 1 has to be a representation of some element in the set. Let us assume that we can order the elements of the set in some way. The binary representation will result in an infinite table, where each row will correspond to an element in the set.

Let us take now the compiment of the diagonal - it is a representation of some element in the set, so it should appear somewhere among the rows of the table. However, it differs form the first row in the first element, it differs from the second row in the second element, etc, and hence it is not equal to any row in the table. This contradicts the assumption that all elements can be ordered, and each element corresponds to a row in the table. Hence the set is uncountable.

Here are some examples of proofs that use the diagonalization principle.