>With infinite number sets, Will I always be able to find a formula to group two sets of infinite numbers?
With the questions that appear on the test, you should always be able to find the formula.
In general the formulas can be very complicated, sometimes so complicated that nobody can figure them out. However we won't ask you anything more complicated than, say, n <--> 5n + 2, where the numbers in one set 'jump' by the same amount each time.
>I know that with infinite set of numbers, when grouping, I can find its corresponding number if they are of one to one ratio. What if they are not a one to one ratio?
If the sets aren't in 1-1 correspondence then you won't be able to find a formula to create one.
>Can I still find a formula for the corresponding numbers of these sets?
>Also, can sets of irrational numbers be in an infinite set? If so, then how?
Any nonrepeating, nonterminating decimal number is irrational. A simple example is .01001000100001 ... . This number has a pattern but it never repeats the same sequence of numbers (there is always 1 more zero with every new segment of the pattern, so it isn't strict repetition).
One of many possible infinite sets of irrational numbers would be
{.010010001..., .020020002..., ..., .0n00n000n ..., ... },
where n goes successively through all counting numbers.
It is not possible to put the irrational numbers into 1-1 correspondence with the counting numbers. This can be proved fairly easily, but it's a little beyond the scope of our course.
>I know that irrational numbers are infinite, but they cannot be represented in fraction form.
>So then, if they too can be grouped in infinite sets, then how would I find a formula for those sets?
It can be proved that there is no possible formula for relating the irrational numbers to the counting numbers. So you can't find a formula to do this.