Show the set of all functions $f : A \rightarrow \{0,1\}$ is equinumerous with the power set of $A.$
23125
Prove that there is a bijection between the set of all subsets of $X$, $P(X)$, and the set of functions from $X$ to $\{0,1\}$.
93102
prove that $f:X\rightarrow Y$ is surjective if and only if $f(f^{-1}(C))=C$
76148
Show that $\exists c \in \mathbb{R} : f(c) = 0$
73850
Cardinality of set of functions
29065
Encode each $n_1,n_2,n_3,...∈N^N$ by an inﬁnite sequence of 0s and 1s with inﬁnitely many 0s, and give a proof that $N^N$ is equinumerous with $R$.
148813
Prove: $f: \mathbb{R} \rightarrow \mathbb{R}$ st for every $x \in \mathbb{R}$ there exists $n$ st $f^{(n)}(x) = 0$, f is a polynomial.
1821
Proving $f(C) \setminus f(D) \subseteq f(C \setminus D)$ and disproving equality
124424
How to show that there is a deduction of $\varphi\leftrightarrow\psi$
141225
Number of all bijective functions from A to A.
113687
$f$ is continuous function from $ [0,1]$ to itself and f(0)=0, f(1)=1 and $f(f(x))=x$ then show that $f(x)=x$ on$[0,1]$
45219
Proving that $C$ is a subset of $f^{-1}[f(C)]$
144735
Possible cardinality of power sets
102529
Prove that the set $\left\{ x, Ax, \dots, A^{k-1} x \right\}$ is linearly independent
46875
The identity morphism in $\mathbf{Set}$ is the identity function
33099
If $f$ is an entire function where every power series expansion has at least one 0 term, show it is a polynomial
23071
Showing that $f_0 (x_1, \ldots, x_m) \mathrm tr A =\displaystyle{ \sum_{i=1}^n} f_0(x_1, \ldots, Ax_i,\ldots, x_m)$
106748
Number of functions $f : A \to A$ such that $f(f(x))=f(x)$