I'd be extremely grateful if anyone could give me a hint on how to create an injective function .
Thanks in advance.
Is the following function defined properly?
We take a sequence of natural numbers, first we change the numbers into their binary representations, then we put before each number, except the last one, where we put a . For example, . We do this so that for every sequence of natural numbers the sequence of 0's and 1's is different, because the 1 on odd position signifies the end of sequence for some natural number. We do this because without using only binary representations would begin the same as .
I am not sure I understand.
Note the set is usually denoted by
Here is a standard theorem: For any infinite set it is the case that . That theorem takes some work to prove.
You need the Schroder-Bernstein theorem, use the characteristic function and the fact that .
However, encoding natural numbers in binary is also easy. One way is to repeat every bit of a number twice and insert 01 between the numbers. For example, 2 5 4 in binary is 10 101 100, so the encoding is 11000111001101110000.
Yes I'm encoding an infinite sequence.
I don't think it matters if the binary representation of a number has even or odd number of digits. I probably wasn't clear on this, but we put the 0's before EVERY digit of a binary representation, not just the first one, and 1 before the last one. If we add a single digit before every digit we will get a string of 0's and 1's that's two times longer than it was before. I flubbed writing my example above, it should be . Other example . In the end the only 1 on an odd position is the one before the last digit in the original representation. I do this, because otherwise the sequences , would begin the same way if I were only putting in the sequence the binary representation of numbers. This way for every possible infinite sequence of natural numbers a unique sequence of 0's and 1's would be provided.