Note that ; from the set the subset

can be represented by the string 001010101. If fact, any subset of four such that no two are consecutive can be represented by a 9-bit string of four 1’s and five zeros in which there are no adjacent 1’s.

Thus, from the set we can use an n-bit string consisting of k 1’s and n-k 0’s, having no adjacent 1’ to represent a subset of k elements containing no consecutive integers.