- #1

lovemake1

- 149

- 1

## Homework Statement

Question asks to prove that all binary string that begins and ends with the same number(bit) has an even number of occurences of substrings from {0,1}

Hint: you may find useful to combine claims about strings that start and end with different bit

## Homework Equations

{0} = {} since 0 is a multiple of 2 it holds.

{11} = {} since 0 is a multiple of 2 it holds.

eg 010010 = (01), (10), (01) (10)

## The Attempt at a Solution

My induction hypothesis was, for all natural number n, assume that p(j) holds for some integer j such that 0 <= j < n, that is a binary string with length j that begins and ends with the same bit has an even number of ooccurences of substrings from {0,1}

so, let j = n - 1 and by I.H we know that p(j) holds.

j, begins and ends with the same bit. and can be represented like so, P(j) = 2t for some integer multiple t. since 1st and last digit of j is the same, we know if n = jlast then there are no additional sets created so the equation will still be p(n) = 2t + 0;

but if we were to add n != jlast, then there would be 1 extra subset at the end since ...01 or ...10 will take place. therefore, the equation will now be p(n) = 2t + 1; even number + odd = odd. hence we've shown that for all n, if first and last digit of string is the same, there are even number of substrings from {0,1}

I feel like I have done too much explaining in english and not enough mathenatical proof.

is there somewhere I went too far? help is appreciated.

Last edited: