What does it mean $|A|<|B|$, when $A$ and $B$ are infinite sets?

Question

For infinite sets, $A$ and $B$, what does it mean to say $|A| < |B|$? Does it mean that there is an injection $A \to B$?

Answer 1

$|A|\le|B|$ means exactly that there is an injection $A\to B$. $|A|<|B|$ means that there is such an injection, but there is no bijection between $A$ and $B$. By the Cantor-Schröder-Bernstein theorem this is equivalent to saying that there is an injection $A\to B$ but no injection $B\to A$.

Answer 2

The definition of $|A|<|B|$ is this:

1. There exists an injection from $A$ into $B$.
2. There does not exist a bijection from $A$ onto $B$.

Equivalently, one can formulate it as:

• There is a subset $B'\subseteq B$ and a bijection from $A$ onto $B'$, but there is no bijection from $A$ onto $B$.

These definition, by the way, has nothing to do with infiniteness or finiteness. It works for sets in general.

For example, $|\Bbb N|<|\Bbb R|$. There exists an injection, $f(n)=n$, for example, but there does not exist a bijection as Cantor's diagonal argument tells us.

Answer 3

In general (in a quasi-ordered set, say) $a<b$ means that $a\le b$ and $b\not\le a$. For cardinal numbers of sets, $|A|\le|B|$ means that there is an injection from $A$ to $B$. Thus, $|A|<|B|$ means that there is an injection from $A$ to $B$ but there is no injection from $B$ to $A$.

In view of the Cantor-Bernstein theorem, it follows that $|A|<|B|$ holds if and only if there is an injection from $A$ to $B$ but there is no bijection between $A$ and $B$. On the other hand, if we're dealing with order types of ordered sets, denoted by $tp(A)$, it would be wrong to think that $tp(A)<tp(B)$ means that $A$ is embeddable in $B$ but not isomorphic to $B$.