Hot answers tagged notation
64
For a real number $x$,
$$\lfloor x\rfloor=\max\{n\in\mathbb{Z}\mid n\leq x\}.$$
I'd like to add though, that "$\lfloor x\rfloor$" is mathematical notation, just as much as the right side of the above equation is; the right side might use more "basic" constructions, but you can then ask about
$$\max,\qquad\in,\qquad \mathbb{Z},\qquad {}\mathbin{\mid}{},\qquad ...
17
It really depends on context. But be safe; just say $x > 0, x\in \mathbb R$.
Omitting the clarification can lead to misunderstanding it. Including the clarification takes up less than a centimeter of space. Benefits of clarifying the domain greatly outweigh the consequences of omitting the clarification.
Besides one might want to know about rationals ...
12
When you have a lot of integrals, particularly with limits, it can be very helpful at times to be able to tell at a glance which integral is over which variable.
$$\int_0^1 \int_2^3 f(x,y) \; \mathrm d x \mathrm d y$$
This is not particularly readable or clear, especially when $f$ is lengthy and there are more nested integrals etc. I could also imagine it ...
12
$\ln 0$ is undefined. Why is that? Remember that $y = \ln x$ is defined as the unique number staisfying $e^y = x$. But we know that the exponential function is always positive, so what happens if we take $x = 0$? Then there's no $y$ that will make the equation $e^y = 0$ true, so $\ln 0$ is undefined.
However, we can take limits, and in that context we can ...
10
From Wikipedia: Surjective funtion
A surjective function is a function whose image is equal to its codomain. Equivalently, a function f with domain $X$ and codomain $Y$ is surjective if for every $y$ in $Y$ there exists at least one $x$ in $X$ with $f(x)=y$. Surjections are sometimes denoted by a two-headed rightwards arrow, as in $f : X ...
8
I like the factorial base,
where the integer part of a real number
is written as
$\sum_{i=2}^n a_i i!$
where the $a_i$ are integers such that $0 \le a_i < i$
and the fractional part is written as
$\sum_{i=2}^{\infty} \frac{b_i}{i!}$
where the $b_i$ are integers such that$0 \le b_i < i$.
The nice thing about this is that
the integer part has a unique ...
7
Assuming the Axiom of Choice, the first cardinal after $\omega=\aleph_0$ is (as Brian notes) $\omega_1$ or $\aleph_1$; it can be defined as $\sup\{\alpha : \alpha\text{ is a countable ordinal}\}$.
Without the Axiom of Choice, however, it's consistent that there is no first cardinal greater than $\aleph_0$; in particular, you can have both $\aleph_1$ and ...
6
While bubba raises valid points about base 2 from a practicality standpoint, I myself would defend the choice of base 2 for the following reason: it makes addition and multiplication incredibly easy. This is, in fact, the way computers do these basic operations.
Addition in binary operates under the following rules:
$0 \land 0 = 0$
$1 \land 0 = 1$
$0 ...
6
Brian Hayes in his American Scientist article Third Base argues that "When base 2 is too small and base 10 is too big, base 3 is just right."
Figure 1 has the caption
Most economical radix for a numbering system is $e$ (about $2.718$) when economy is measured as the product of the radix and the width, or number of digits, needed to express a given range ...
6
Probability distribution and probability measure are synonyms.
$[X=a]=X^{-1}(\{a\})=\{\omega\in\Omega\mid X(\omega)=a\}$ hence $P(X=a)=P(X^{-1}(\{a\}))$.
The distribution of the random variable $X:\Omega\to\mathbb R$ is the unique probability measure $\mu$ on $\mathcal B(\mathbb R)$ defined by $\mu(B)=P(X\in B)$ for every $B$ in $\mathcal B(\mathbb R)$, ...
5
Balanced Nonary (base 9) would probably be really good. The digits go from -4 to 4, so taking the negative of a number would just be taking the negative of each digit, so subtraction is easy. Multiplication and division are particularly easy too if you make the easy conversion to balanced ternary first. Then there's no carrying when multiplying single digits ...
5
The term almost certainly refers to the product space $\omega^\omega$, the product of countably infinitely many copies of the countably infinite discrete space, which is well-known to be homeomorphic to the space of irrational numbers with the usual topology. More generally, if $D_\kappa$ is the discrete space of cardinality $\kappa$, $D_\kappa^\omega$ is ...
5
One way to rationalise this notation is to think of a proposition as having a certain information content.
Then $p \supset q$ can be thought of as "the information content of $q$ is contained in that of $p$". A particular piece of information that can be obtained from $q$ is "$q$ is true".
Thus we see that $p \supset q$ naturally gives rise to the ...
5
Actually, historically, the reason we use $\supset$ is that Peano originally wrote $p C q$ for "$p$ is a consequence of $q$", and wrote a backward "$C$" for "$p$ has as a consequence $q$". Eventually, just as the "$\epsilon$" became "$\in$", so too did the backward "$C$" become "$\supset$". So it doesn't actually have anything to do with set theory, per se, ...
4
The key fact here is that you are confusing the two types of multiplications. The first is the multiplication of a matrix by a scalar which I will denote using $\cdot$. The second is matrix-matrix multiplication which I will denote as $\times$.
Your equation is
$$(\mathbf{u}^\mathrm{T}\times \mathbf{v}) \cdot \mathbf{v}$$
There is no reason to expect the ...
4
For computer applications, bases like 2, 8 and 16 are obviously the best. Given that a large percentage of numerical data is stored in and processed by computers, these days, one could argue that what's good for computers is good for society.
Of the three I mentioned, I suppose that 8 or 16 would be better than base 2. Having the price of bananas as a ...
4
Of course, everything depends on context. I usually prefer to say things like
For any $x\in\mathbb{R}$ with $x>0$, ...
instead of
For any $x>0$, ...
to remove ambiguity, but I'm not insistent on it; I might be willing to sacrifice the "$x\in\mathbb{R}$" if it's making an orphan at the end of a paragraph, for example.
In contrast, everyone ...
4
There's no such thing as one "official mathematical notation". There are a bunch of basic conventions that most authors follows, and each field has its own special ways of expressing things. The best advice is usually to follow what other people in your field are doing. I.e., if there are publications which deal with algorithms similar to the one you've ...
4
Using the second part of the definition recursively, we see $S$ must contain
$\varnothing=0$
$0\cup\{0\}=1$
$1\cup\{1\}=2$
$2\cup\{2\}=3$
$\cdots$
which are all distinct. Note that when we look at the set-theoretic construction of the naturals, we have $0=\varnothing$ and $n=\{k:k<n\}=\{k<n-1\}\cup\{n-1\}=n-1\cup\{n-1\}$.
4
Yes, those entries: $m_{ij}$ in $M$, with $i = j$, constitute what can technically be called the "main diagonal" of the rectangular matrix, though the diagonal of such a matrix is not necessarily as "useful" as it is in a square matrix. See, e.g., the Main Diagonal entry in Wikipedia.
4
I suspect that this question can be better articulated as: how can we compute the floor of a given number using real number field operations, rather than by exploiting the printed notation, which separates the real and fractional part, making nearby integers instantly identifiable.
How about as Fourier series? If we subtract the function $y = x$ from ...
4
$$Φ^{-1}\left[\{Φ(g)\}\right]=gH=Hg\tag{1}$$
$(1)$ simply denotes the preimage (the inverse mapping) of the image of the "set": $\{\Phi(g)\} \subseteq G/H$ under $\Phi$, assuming in this case we have that $H$ is normal. Since the "object" being mapped by the homomorphism $\Phi$ is technically a set, instead of enclosing the set in parentheses, as we would ...
3
Some use the notation $f^{-1}[X]$ to denote the preimage of a set $X$ under a map $f:A\to B\supseteq X$: apparently the brackets prevent ambiguity in set theory, as one of my instructors explained to me.
Here is an example justifying the need. Suppose we build the natural numbers $\bf N$ as ordinals:
$0:=\{\}$
$1:=\{0\}=\{\{\}\}$
...
3
1) and 2) are explained in page 8, ${\scr P} x$ indicates that $x$ satisfies some predicate or some property (for example consider $x$ satisfies $\scr P$ if and only if $x$ is a positive integer then $2,23$ does not satisfy $\scr P$ and therefore is not an element of $\{x:{\scr P} x\}$), and $a _x$ is a function of $x$, so you can just substitute that by ...
3
$$\lfloor x \rfloor = \text{supremum} \{n \in \mathbb{Z} : n \leq x\}$$
For a subset $S \subset X$, where we have an order $\leq$ on $X$, a supremum or least upper bound of $S$ is an element $M \in X$ such that
$$x \leq M, \,\,\,\,\, \forall x \in S$$
and for any $m \in X$ such that $x \leq m$ for all $x \in S$, we have $M \leq m$.
3
Several inconsistencies, which might hinder your understanding:
Let X(w) be a real random variable on ($\Omega$ , P).
No, a real random variable is a mapping $X:\Omega\to\mathbb R$, not the image $X(\omega)$ of a specific element $\omega$ of $\Omega$ by the mapping $X$.
The image X($\Omega$) the set of all the values X(w) can take ,written ...
3
It is the set of all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ that are continuous in every $x\in\mathbb{R}$.
It would be best to denote it as $C^0(\mathbb{R})$, motivated by the definition
$$
C^r(\mathbb{R})=\{f:\mathbb{R}\rightarrow\mathbb{R}|f \text{ is of class }C^r\}
$$
for $r\geq 0$.
3
Well, since parentheses exist precisely to specify the intended order of operations in case the usual default rules don't cut it, it makes sense that they come first
As for exponentation, I'd say that this is a consequence of using superscripts to indicate exponentation, since those (via font size) provide a natural grouping. It'd certainly be very weird if ...
3
Let's begin by understanding the letters and their meanings, then we'll give context to everything.
$\exists$ is a quantifier. It means that the following symbol is a variable (a set, in the case of set theory) and we assert there is an object which the properties which we require that symbol to have.
$S$ is that symbol. It is a placeholder that will be ...
3
The colons are ordinarily used for homogeneous coordinates. For example, consider the real plane. We have Cartesian coordinates $(x,y)$. The homogeneous coordinates $(x:y)$ are equivalence classes of points. We say that two, non-zero points $(x_1,y_1)$ and $(x_2,y_2)$ are equivalent if and only if there exists a non-zero real number, say $\lambda$, for which ...
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