How can you define equivalence of infinite sets?
For example, is the set of positive integers equal to the set of positive even integers? Can we go so far as to just say all infinite sets are equivalent?
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First you need to be careful with the word equivalence. Equivalent sets have the same members, loosely speaking, so the integers and the positive integers are not equivalent.
However, the "size" of the integers is the same as the "size" of the positive integers. Here we are comparing their cardinality. Two sets have the same cardinality if there is a one-to-one correspondence between the members of the sets; in other words you can match every element of one set to an element in the other set.
(1) The set of positive integers has the same cardinality as the set of positive even integers. If A is the set of positive integers, and B the set of positive even integers, then for every a in A we find b=2a in B. Thus every element in A is matched with an element 2a in B.
(2) Not all infinite sets have the same cardinality. Amazingly there are the same number of elements in the positive integers, the integers, and even the rational numbers. (See Cantor's diagonalization argument). But there are more real numbers than counting numbers. (We say a set is countable if its cardinality is the same as the natural numbers -- integers, rationals are countable, reals are not)
We can use power sets to create sets of larger and larger cardinality. The set of integers has cardinality aleph-nought `aleph_0` , while the reals have cardinality `2^(aleph_0)` . Sets can have larger cardinality, but it is an open question whether there is a set such that its cardinality is `aleph_0<c<2^(aleph_0)` ; Cantor supposes that there is no such set and this is known as the continuum hypothesis.
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