Intereting Posts

How should I calculate $\lim_{n\rightarrow \infty} \frac{1^n+2^n+3^n+…+n^n}{n^n}$
Example in which a normal subgroup acts non-equivalent on its orbits
Relationship between very ample divisors and hyperplane sections
Evaluate $\sum\limits_{k=1}^n k^2$ and $\sum\limits_{k=1}^n k(k+1)$ combinatorially
Number of reflexive relations defined on a set A with n elements
transcendental entire function, $Aut(\mathbb{C})$
Why do this algorithm for finding an equation whose roots are cubes of the roots of the given equation works?
How to prove this inequality(7)?
Rouché's Theorem for $p(z)=z^7-5z^3+12$
Why does the strategy-stealing argument for tic-tac-toe work?
Uniform Integrability
What's wrong with this argument? (Limits)
What Is a Morphism?
Mathematical expression to form a vector from diagonal elements
Partitions of the odd integers

So I was solving a problem in Rudin (chapter 3 #16, to be specific) and I realized how convenient it would be to have a symbol that represented an undetermined equivalence relationship. As an example I will use the symbol $\sim$.

$\mathbf{Example}$. Suppose we have an expression $A$ that we want to relate to $B$. We then set

$$

A \sim B

$$ and perform an algebraic operation to obtain $A’$ and $B’$. If this operation involved multiplying both sides by a negative number, we change $\sim$ to $\sim’$. So suppose it did; then we have

$$

A’ \sim’ B’.

$$ Lets perform another algebraic operation, and suppose again that we multiplied by a negative number. Then $\sim’$ becomes $\sim$ (since inequalities are the same under two multiplications by negatives), and we obtain $A”$ and $B”$ and thus

$$

A” \sim B”.

$$ We can continue on in this fashion until we have have done $n$ operations. For simplicity let us suppose that at this point our equivalence relation is $\sim’$. Then

$$

A^{(n)} \sim’ B^{(n)}.

$$ Suppose further that we actually know that $A^{(n)} < B^{(n)}$. Then we can conclude

$$

A > B.

$$ If instead we knew that $A^{(n)} = B^{(n)}$ we would have $A = B$, and if instead we knew that $A^{(n)} > B^{(n)}$ we would have $A < B$.

Does anyone know of a symbol such as this, and if so, are there interesting things to be said about essentially solving for equivalence relations?

- The class of equivalence.
- Equivalence relations on natural numbers
- Equivalence Relations On a Set of All Functions From $\mathbb{Z} to $\mathbb{Z}$
- Equivalence relations on classes instead of sets
- Is this alternative definition of 'equivalence relation' well-known? useful? used?
- When is the topological closure of an equivalence relation automatically an equivalence relation?

- What is the proper notation for integer polynomials: $\Bbb Y=\{p\in\Bbb Q\mid p:\Bbb Z\to \Bbb Z\}$?
- What are $\Sigma _n^i$, $\Pi _n^i$ and $\Delta _n^i$?
- What is “multiplication by juxtaposition”?
- Equivalence Relation problem
- Is there a name for this property: If $a\sim b$ and $c\sim b$ then $a\sim c$?
- The notation for partial derivatives
- What does a binomial coefficient with commas mean?
- I don't understand equivalence relations
- Question about set notation: what does $]a,b[$ mean?
- Difference between $:=$ and $=$

If I’m not sure whether the relationship is $<$, $=$, or $>$, I usually use a circled question mark. In LaTeX, type `\textcircled{\scriptsize ?}`

.

I can’t seem to do this in MathJax; perhaps the closest would be to type `$\fbox{?}$`

to get $\fbox{?}$.

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- Prove or disprove: $\sum a_n$ convergent, where $a_n=2\sqrt{n}-\sqrt{n-1}-\sqrt{n+1}$.
- Are these exactly the abelian groups?
- What's an example of a number that is neither rational nor irrational?
- Showing that $\lim_{n\to\infty}\sum^n_{k=1}\frac{1}{k}-\ln(n)=0.5772\ldots$
- What we can say about $-\sqrt{2}^{-\sqrt{2}^{-\sqrt{2}^\ldots}}$?
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- Product and Box Topologies
- Why all differentials are $0$ for Serre Spectral Sequence of trivial fibration?
- Prove that $\frac{\binom{p}{k}}{p}$ is integral for $k\in \{1,..,p-1\}$ with $p$ a prime number with a particular method
- Euler's Solution of Seven Bridges of Königsberg in Layman Terms
- Using expected value to prove that there is a line intersecting at least 4 of the circles
- Incoherence using Euler's formula
- Infinite sum of floor functions