Are all infinite sets equal according to one to one correspondence?

Answer 1

No.

The "smallest" infinity is the cardinality of the natural numbers, N. This cardinality is named Aleph-null. Rational numbers also have the same cardinality as do n-tuples of rational numbers.

The next larger cardinality is that of the real numbers. This is the "continuum, C, which equals 2aleph-null. As with the cardinality of the natural numbers, n-tuples of reals have the same cardinality.

The point about introducing n-tuples, is that they are used to denote points in n-dimensional space.

If you want more read some of the Wikipedia articles of Cantor, Hilbert's Grand Hotel. These could lead you to many more related articles - though sadly, not infinitely many!

Answer 2

No, all infinite sets are not necessarily equal according to one-to-one correspondence. One-to-one correspondence is a way to compare and classify infinite sets based on their cardinality. Sets that have a one-to-one correspondence are said to have the same cardinality, which means they are equal in size. However, not all sets have the same cardinality. For example, the set of natural numbers (countably infinite) has a different cardinality than the set of real numbers (uncountably infinite).