Any two sets $Y$, $Z$ have the same cardinality $\iff$ there are injective functions $f: Y \rightarrow Z$ and $g: Z \rightarrow Y$.
133699
Proving that there is a bijection between $A$ and $B$.
13273
Cardinality of the set of all real functions of real variable
21663
Bijection between power sets of sets implies bijection between sets?
22232
Proving that cardinality of the reals = cardinality of $[0,1]$
9082
Do sets, whose power sets have the same cardinality, have the same cardinality?
81509
The cardinality of a countable union of countable sets, without the axiom of choice
92639
Prove that the family of open sets in $\mathbb{R}$ has cardinality equal to $2^{\aleph_0}$
98518
Cardinality of equivalence relations on N
150790
Prove two sets have same Cardinality by writing down bijection
89712
For every set $A$, the empty set is a subset of $A$. The empty set is a set. Therefore, the empty set has a cardinality $\geq 1\ldots$
15733
Show that the cardinality of $\mathbb N$ is the same as the cardinality of $\mathbb N \times \mathbb N$
122947
What is the cardinality of the set of infinite cardinalities?
28223
Prove/Disprove that if two sets have the same power set then they are the same set
96845
Question: Compare the cardinality between the sets $\Bbb R$ and $\Bbb N^{\Bbb N}$
144735
Possible cardinality of power sets
92376
Let $f:A \rightarrow B$ and C ⊂A. Define f[C]={b∈B: b=f(a) for some a∈C}. Prove or disprove each of the statement
142267
If two sets have the same cardinality, then so do their power sets. Converse can't be answered?