# Do all infinite sets have the same cardinality?

### Do all infinite sets have the same cardinality?

No. There are cardinalities strictly greater than |N|.

### How do you prove two infinite sets have the same cardinality?

If this is possible, i.e. if there is a bijective function h : A → B, we say that A and B are of the same cardinality and denote this fact by |A| = |B|. If two (finite or infinite) sets A and B are not of the same cardinality, we can try to compare which one of the two sets has at least as many elements as the other.

### Can infinite sets be equal sets?

Although we have not defined the terms yet, we will see that one thing that will distinguish an infinite set from a finite set is that an infinite set can be equivalent to one of its proper subsets, whereas a finite set cannot be equivalent to one of its proper subsets.

### What is the difference between countably infinite and infinite?

For example, the set of integers {0,1,−1,2,−2,3,−3,…} is clearly infinite. ... But to stress that we are excluding finite sets, we usually use the term countably infinite. Countably infinite is in contrast to uncountable, which describes a set that is so large, it cannot be counted even if we kept counting forever.

### Are all countably infinite sets Equicardinal?

Having stated the definitions as above, the definition of countability of a set is as follow: Definition 3.6 A set E is said to be countably infinite if E and N are equicardinal. And, a set is said to be countable if it is either finite or countably infinite. The following are some examples of countable sets: 1.

### How do you prove an infinite set?

A set is infinite if and only if for every natural number, the set has a subset whose cardinality is that natural number. If the axiom of choice holds, then a set is infinite if and only if it includes a countable infinite subset. If a set of sets is infinite or contains an infinite element, then its union is infinite.

### Are any two infinite sets equivalent?

The definition we introduced for countable sets can now be restated as: a set is said to be countable if it is equivalent to the set of natural numbers. Theorem 1. Every infinite set is equivalent to some proper subset of itself.

### Can a set be infinite?

An infinite set is a set whose elements can not be counted. An infinite set is one that has no last element. An infinite set is a set that can be placed into a one-to-one correspondence with a proper subset of itself.

### Is Z countably infinite?

The set Z of integers is countably infinite.