Intereting Posts

Miller-Rabin-Test : Why can we be $100\%$ certain after $4$ tests for $p=13$?
When is a function satisfying the Cauchy-Riemann equations holomorphic?
Name for collection of sets whose intersection is empty but where sets are not necessarily pairwise disjoint
Prove that:$\int_{0}^{1}{x+x^2+\cdots+x^{2n}-2nx\over (1+x)\ln{x}}dx=\ln{\left}$
The Hairy ball theorem and Möbius transformations
Modular Forms: Find a set of representatives for the cusps of $\Gamma_0(4)$
Show that a matrix $A=\pmatrix{a&b\\c&d}$ satisfies $A^2-(a+d)A+(ad-bc)I=O$
How do I simplify and evaluate the limit of $(\sqrt x – 1)/(\sqrt x – 1)$ as $x\to 1$?
Is there limit $ \lim_{(x,y) \to (0,0)} \frac{x^3}{x^2 + y^2}$?
How is $\mathbb N$ actually defined?
How to get a reflection vector?
If a holomorphic function $f$ has modulus $1$ on the unit circle, why does $f(z_0)=0$ for some $z_0$ in the disk?
Are the unit partial quotients of $\pi, \log(2), \zeta(3) $ and other constants $all$ governed by $H=0.415\dots$?
Finding a basis for a certain vector space of periodic polynomials
Proof of strictly increasing nature of $y(x)=x^{x^{x^{\ldots}}}$ on $[1,e^{\frac{1}{e}})$?

Let us consider a set $A$. let $B$ be an element of the set. Now what I want to know is that whether saying $B$ is contained in $A$ and $B$ belongs to $A$ means the same? Could anyone here cite any context where they do not mean the same?

- Examples of transfinite induction
- Show that the set of all infinite subsets of $\mathbb{N}$ has cardinality greater than $\mathbb{N}$
- Is the Cartesian product of sets associative?
- Cardinality of the set of all (real) discontinuous functions
- Show that $A \cap B = B$ iff $A \cup B = A$, where $A \subseteq B$.
- Cardinality of the union of infinite and countable sets
- If $g \circ f$ is the identity function, then which of $f$ and $g$ is onto and which is one-to-one?
- Proof that the real numbers are countable: Help with why this is wrong
- Constructing a bijection from (0,1) to the irrationals in (0,1)
- $\exists\text{ set }X:X=X^X$?

Paul Halmos in his autobiography reports that he once decided that henceforward he would say “$x$ contains $y$” when he meant “$y$ is a member of $x$” and “$x$ includes $y$” when he meant “$y$ is a subset of $x$”. He adhered to that usage fastidiously for 18 months. At the end of that time he drew his conclusions: (1) the practice is harmless, and (2) he didn’t think anybody ever noticed.

I was inclined to agree with the usage on the grounds that people speak of a family of subsets being “partially ordered by inclusion” but they never say “partially ordered by containment” as far as I know.

And as far as I know, “$x$ belongs to $y$” would meant the same thing as “$x$ is a member of $y$”.

But *sometimes* people say “$x$ is contained in $y$” when they mean $x$ is a subset of $y$. And sometimes they say the same thing when they mean $x$ is a member of $y$. So always make it clear which meaning you have in mind. Sometimes context is enough for that and probably sometimes it is not.

- What are some natural arithmetical statements independent of ZFC?
- What is the purpose of Jordan Canonical Form?
- Prime Appearances in Fibonacci Number Factorizations
- Proximal mapping of $f(U) = -\log \det(U)$
- Countable ordinals are embeddable in the rationals $\Bbb Q$ — proofs and their use of AC
- How to write special set notation by hand?
- Find the form of a second linear independent solution when the two roots of indicial equation are different by a integer
- Understanding the quotient ring $\mathbb{R}/(x^3)$.
- Show that any solution of second order differential equation has atmost a countable number of zeroes $?$
- Give an algorithm that computes a fair driving schedule for all people in a carpool over $d$ days
- How to calculate discrepancy of a sequence
- If $A=AB-BA$, is $A$ nilpotent?
- Proof for power functions
- Limits of recurrently defined sequences.
- Fundamental solution to Laplace equation on arbitrary Riemann surfaces