4:26 PM
in Calculus and analysis, 28 mins ago, by Little Rookie
And there are also instances where i cannot find an explicit expression for the function, then i would suggest a method/algorithm to map the natural numbers to my set
in Calculus and analysis, 27 mins ago, by Little Rookie
sometimes, i show it for first 10 natural numbers, then i would write repeating the process for the other natural numbers
in Calculus and analysis, 27 mins ago, by Little Rookie
Now, the big issue i have is to show that the mapping/function is indeed bijective.
in Calculus and analysis, 27 mins ago, by Little Rookie
In cases, where i do not have the explicit formula for the function, how do i show that it is a bijection?
in Calculus and analysis, 26 mins ago, by Little Rookie
For example, the well known case for Rational numbers, the mapping has no explicit form, instead a counting algorithm is proposed.
4:44 PM
To show this, just notice that if you have a rational $\frac pq$, they are at most finitely many numbers which might have been used in the algorithm before this number.
BTW you do not always have to show that there is a bijection. If you know Cantor-Bernstein theorem then you know that injectiontive functions in both directions are enough.
In the particular case of rationals you could show that $$|\mathbb N| \le |\mathbb Q| \le |\mathbb Z\times\mathbb Z| = |\mathbb N|.$$
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