Intereting Posts

Is there any matrix notation the summation of a vector?
How to check the real analyticity of a function?
Image under an entire function.
Is it necessarly abelian $2$ group?
Find conjugation invariant functions without using eigenvalues?
Conic by three points and two tangent lines
Representing a real valued function as a sum of odd and even functions
Complex differentiability vs differentiability in $\mathbb{R}^2$
Are there simple models of Euclid's postulates that violate Pasch's theorem or Pasch's axiom?
Fundamental Theorem of Algebra for fields other than $\Bbb{C}$, or how much does the Fundamental Theorem of Algebra depend on topology and analysis?
How to use LU decomposition to solve Ax = b
Reflected rays /lines bouncing in a circle?
How discontinuous can a derivative be?
The vertices of an equilateral triangle are shrinking towards each other
Limit of this series: $\lim_{n\to\infty} \sum^n_{k=0} \frac{k+1}{3^k}$?

For each formula $\phi$ without $Y$ free, the universal closure of the following is an axiom:

$\forall x\in A \exists !y \phi(x,y) \Longrightarrow \exists Y \forall x\in A \exists y\in Y \phi(x,y)$

My question is about the $!$, does that mean that $y$ is bound in $\phi$? What does the exclamation mark mean? Is it there so that $y\ne \{x: x\not\in x \}$? I am confused.

- Can there exist an injective function from $\mathbb R$ to $(0,1)$?
- Produce an explicit bijection between rationals and naturals?
- Intersect and Union of transitive relations
- Is $\operatorname{card}(I)=\operatorname{card}(D)$
- Recursive Mapping
- Bijection between natural numbers $\mathbb{N}$ and natural plane $\mathbb{N} \times \mathbb{N}$

- what is the relationship between ZFC and first-order logic?
- Can one come to prove Cantor's theorem (existence of higher degree of infinities) FROM Russell's paradox?
- How to Understand the Definition of Cardinal Exponentiation
- Why does the empty set have a cardinality of zero?
- Prisoners Problem
- I need to disprove an alternate definition of an ordered pair. Why is $\langle a,b\rangle = \{a,\{b\}\}$ incorrect?
- Cantor's First Proof that $\Bbb{R}$ is uncountable
- Namesake of Cantor's diagonal argument
- Ordered pairs in a power set
- Logical Form - Union of a Set containing the Power Set with Predicate/Propositional Function

Rustyn, $\exists !$ is the quantifier “there is a unique”. This is just an abbreviation, as it can be defined in terms of the standard quantifiers: $\exists! x\psi(x)$ is just $\exists x(\psi(x)\land\forall y(\psi(y)\to y=x))$.

Anyway, this version of replacement (where the graph of $\phi$ is a “function”) is equivalent to the more generous version where the $\exists!$ is replaced by the usual $\exists$ (where the graph of $\phi$ is just a “relation”).

- Topology of a cone of $\mathbb R\mathbb P^2$.
- Show that for any prime $ p $, there are integers $ x $ and $ y $ such that $ p|(x^{2} + y^{2} + 1) $.
- Suppose $f : \mapsto R$ is continuous, $f$ is differentiable on $(0,1) $and $f(0)=0$.
- Is this proof correct for : Does $F(A)\cap F(B)\subseteq F(A\cap B) $ for all functions $F$?
- Find the unique pair of real numbers $(x,y)$ that satisfy $P(x)Q(y)=28$
- Find some kind of generating functions for odd Hermite polynomials
- Quotient ring of Gaussian integers
- Prove $\lim\limits_{n\to\infty}\frac{1}{\sqrt{n!}}=0$
- In (relatively) simple words: What is an inverse limit?
- Deriving the inverse of $\mathbf{I}$+Idempotent matrix
- Maximum area of a triangle
- How does $e^{i x}$ produce rotation around the imaginary unit circle?
- Let$\ p_n$ be the$\ n$-th prime. Can you give me a proof for$\ \prod_{i=1}^\infty \frac{p_i-1}{p_i}=P\approx \frac{1}{11.0453}$?
- Let $x_n$ be the (unique) root of $\Delta f_n(x)=0$. Then $\Delta x_n\to 1$
- Prove that $\int\limits_0^1 x^a(1-x)^{-1}\ln x \,dx = -\sum\limits_{n=1}^\infty \frac{1}{(n+a)^2}$