If the number of functions from $A \to B$ is equal to $A$, then the cardinality of A and B is $1$.
136835
ZF Set Theory - If $|A|^{|B|} = |B|$, then $|A| = 1 = |B|.$
21726
Show that if $(b,c)=1$, then $(a,bc)=(a,b)(a,c)$.
15733
Show that the cardinality of $\mathbb N$ is the same as the cardinality of $\mathbb N \times \mathbb N$
47135
Prove if a = b, then f(a) = f(b) for any function f (with natural deduction)
65953
Cardinality of the set of differentiable functions
83518
Proving Two sets have same cardinality
21431
The set of all functions from $\mathbb{N} \to \{0, 1\}$ is uncountable?
55924
How many distinct functions can be defined from set A to B?
20878
If $(a,b)=1$ then prove $(a+b, ab)=1$.
79974
The cardinality of Lebesgue sets
52787
If $\bigcup F = A$ then $A\in F$ then prove that $A$ has exactly one element
13398
Prove that the interval $(0, 1)$ and the Cartesian product $(0, 1) \times (0, 1)$ have the same cardinality
7416
Let $ A $, $ B $ and $ C $ be sets. If $ A \cup B = A \cup C $ and $ A \cap B = A \cap C $, then show that $ B = C $.
63183
Suppose $a,b,m,n$ are natural numbers, prove that if $a({m \over n})=b$ and $(a,n)=1$, then ${m \over n} \in \mathbb{N}$
109745
If $(a,k)=(b,m)=1$, prove that $(ab,km)=(a,m)(b,k)$.
13273
Cardinality of the set of all real functions of real variable
50939
If $A$ and $B$ are nonempty sets, prove that $A \times B = B \times A$ if and only if $A = B$