I apologize for not knowing how to make the acutal symbols.  I believe the turnstile means "proves", so I'm confused what it means when it doesn't have anything before it.   Also, what does '"triangle" union {A}' mean? I believe the triangle means difference between sets, so that expression makes no sense to me.
11048
I'm confused about the use of symbols $\vdash$ and  $ \models$. Reading the answers to  and  I see that:  the turnstile symbol $ \vdash $ denotes syntactic implication. Then $ S \vdash \psi$ means that $\psi$ can be derived from the formulae in $S$  the double turnstile $ \models $, denotes semantic implication. Then the notation $A \models B$ means simply that B is true in every model in which A is true.  So I understand that, e.g., $(ZFC+ A) \vdash B$ means that $A \Rightarrow B$ is a theorem in ZFC and, in this sense $\vdash$ is a mathematical symbol for a theorem. While $(ZFC+ A) \models B$ means that $ B$ is true in every model of ZFC in which $A$ is true, and, in this sense $\models$ is not a mathematical but a metamathematical symbol.  I understand correctly? And those are the only uses of the two symbols?
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Hello.  I'm looking for the English name of these two lines in a two dimensional plane:   they go through the origin they make angles of 45° and 135° with the $x$-axis, dividing the plane in two parts    From my native language, they would translate to "first bissector" and "second bissector" or "first diagonal" and "second diagonal", but I could not find any reference to such names.
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I'm studying a course on probability and statistics and at some point this symbol comes up without introduction. It looks like the number one, but slightly bigger and with a double vertical line.  First time it comes up is when discussing stochastic/random variables that are neither continuous nor discreet in an example:   And somewhat later in a proof:    Anyone got an idea what this symbol represents?  EDIT: Thanks for the fast answers, I was still editing the question for better (larger) images :)
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I saw two less than signs on this Wikipedia article and I was wonder what they meant mathematically.    EDIT: It looks like this can use TeX commands. So I think this is the symbol: $\ll$
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I'm reading a  by Andrew Granville on analytic number theory.  On page 22 of the paper, there appears a strange looking symbol, undefined.  I've circled it in red in the screenshot below.    Since it's not defined in the paper, I'm assuming it must be standard notation.   From the context, I'm assuming it means something like "as compared to", or "with reference to", but that's just a guess.  Can anyone identify the symbol, even better explain what it means and/or provide a reference?  Is there a name to speak the symbol?  Thanks in advance.
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I've been in a lengthy discussion today about how to interpret the division symbol. There seem to be two views on what it means.  • Everything on the left hand symbol is divided by everything on the right hand of the symbol. or  • The operator only deals with the term to its immediate left and right.  Which is correct? I tried to find a specific definition for the usage of the division operator but I couldn't find it.
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In either case, one coin flip resulted in a head and the other resulted in a tail. Why is {H,T} a different outcome than {T,H}? Is this simply how we've defined an "outcome" in probability?   My main problem with {H,T} being a different outcome than {T,H} is that we apply binomial coefficients (i.e. we count subsets of sets) in some common probability problems. But if we take {H,T} and {T,H} to be different outcomes, then our "sets" are ordered, but sets are by definition unordered...   I feel as though the fact that I'm confused about something so basic means that I am missing something fundamental. Any help or insight whatsoever is greatly appreciated!
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Is there mathematical sign that combines multiplication ($\times$) and division ($\div$) operations? Would that ever be used? For example $4({\times}{\div})2$ would give $8$ and $2$, like $4\pm 2$ would give $6$ and $2$.
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I am not very well versed on set theory or syntax, but I thought I knew the basics.  However, in a book about databases I am reading now, the author uses $2^x$ to signify "a set of $x$."  For example, $2^{\text{dogs}}$ is a set of $\text {dogs}$, etc.    The author never really explained this or why he does it, I just picked up the meaning from context.    I am not sure why the exponent operator is used, nor am I sure what the number $2$ has to do with it.  The sets being represented are NOT powers of $2$ (in size)... they come in all sizes.  Is this a valid notation?  I have not seen it anywhere before...
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My son is in 2nd grade.  His math teacher gave the class a quiz, and one question was this:     If a triangle has 3 sides, and a rectangle has 4 sides,    how many sides does a circle have?   My first reaction was "0" or "undefined".  But my son wrote "$\infty$" which I think is a reasonable answer.  However, it was marked wrong with the comment, "the answer is 1".  Is there an accepted correct answer in geometry?  edit: I ran into this teacher recently and mentioned this quiz problem.  She said she thought my son had written "8." She didn't know that a sideways "8" means infinity.
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Context: I was looking at . I didn't know the result referred to in Brian M. Scott's answer is hard enough to require a separate proof   ― .  ― .  (1)  ― What's the intuition? Is Bezout Identity supposed to be intuitive?  (2) — Why doesn't this try work? Let $c$ be any common divisor of $a, b$. Therefore $c\mid a, c\mid b \iff cj_1 = a, cj_2 = b$ for some integers $j_1, j_2$.     Call $g = \gcd$. By definition $\gcd(a,b)\mid a$ and $\gcd(a,b)\mid b$ and $c \le \gcd(a,b)$. Therefore $g\mid a, g\mid b \iff gk_1 = a, gk_2 = b$ for some integers $k_1, k_2$.   The two paragraphs together precipitate $cj_1 = a = gk_1,  cj_2 = b = gk_2$ but this only proves $c\mid gk_1, c\mid gk_2$?    Origin - Elementary Number Theory, Jones,  p9, Exercise 1.8
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This problem is from Discrete Mathematics and its Applications  My question is on 9b. I know that the sign represents an element is a member of. (from book)  I know that the O with a slash across it is the empty set which &quot;is a special set that has no elements&quot;. From , I got that the empty set is a subset of all sets, meaning that every member of the empty set(nothing) is also a member of any other set. Based on all of this, for 9b, would {0} contain the empty set because it fundamentally has the elements that consist of the empty set(nothing) or does it physically have to have the empty set?
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I am really lost trying to understand the difference between the logical connective "implies", $\to$, and the metalogical symbol (or maybe it's also a connective?) $\vdash$. (This is all focusing on prepositional logic here).  In metalogical terms, for example with modus ponens, it is said that $P, P \to Q \vdash Q$, which means "If we have a proof of $P$, and we have a proof of $P \to Q$, then we can infer / make a proof of $Q$". But I don't understand what the difference is between that and saying something like $P \land (P \to Q) \to Q$ which is similar but uses $\to$ instead of $\vdash$.  For example the $P \to Q$, at least in my experience, means "it is possible to go from $P$ to $Q$" but I don't see how "going to $Q$" is different from "inferring $Q$." Simply telling me that one is metalogical and one is not doesn't really help me understand what's going on.   I've also been given the example of  but I don't understand this, either. It sounds like the tortoise is constantly rejecting implications because "who says I have to accept conclusions just because the premises are true?" but then somehow introducing a metalogical $\vdash$ solves this? "We use the metalogical symbol $\vdash$ to basically force that stubborn tortoise to accept the conclusions and we've now circumvented the issue."  Unless I've grossly misunderstood something I just don't see why that's even a thing. Who says then I have to accept $\vdash$? Is $\vdash$ just a stronger form of $\to$, like a "sudo $\to$" or something (to borrow a Linux term), a form of implies that forces the conclusion to be accepted from the premise(s)?   What's the difference? How are they working here? Why do we need them? Are there any concrete examples showing the difference of both?
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In the following question  the notation $c\in ]a,b[$ is used. What does this mean? I have never seen it before.
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I see this notation a lot, but I can't seem to find an answer in my google searches. In particular I am looking at the question      Let $\alpha:S^n\rightarrow S^n$ be the antipodal map. Prove that $n$ is odd, then $\alpha\simeq \operatorname{id}.$    If it is an arbitrary symmetric group, then I understand, but I have seen $S^1$ be used for the circle.
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Where does the name of these two famous types of sequences come from?  The article  of Wikipedia says that the geometric sequence is called like this because every term is the geometric mean of its two adjacent terms. Though it is true, it only reduces the question to: why is the geometric mean geometric (in opposition to the arithmetic mean).   Continuing my investigation on , I was told that a square with the same area than a rectangle with sides $a$ and $b$ has their geometric mean $\sqrt{ab}$ for side. That's again totally true, but a square with same perimeter than this rectangle of sides $a$ and $b$ has their arithmetic mean $\frac{a+b}{2}$ for side!  Thus, my question: Who coined these names? And why? Why is the geometric mean more geometric than the arithmetic mean?
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I've been googleing and ran across  little blog post. It has a bit of information stating the the &amp; symbol was at one point the 27th letter in the alphabet.     For years the &amp; symbol (now known as the ampersand) was the final,   pronounced “and” but recited with the Latin “per se,” meaning “by   itself.” The position and pronunciation eventually ran together, with   “X, Y, Z, and per se and” becoming “X, Y, Z, ampersand.”   I have to wonder how could a symbol be in the alphabet? My understanding is that letters represent sounds and the putting the sounds together make words. If &amp; was a letter for the combined sound that the word "and" makes was it ever used in the spelling of a word? Something along the lines of “h&amp;” for hand? Also, why was &amp; in the alphabet and not other symbols like % or #?  I don't think this is a duplicate as I'm not looking for the definition of the &amp; symbol or the story of where it came from but rather its use as a letter in the alphabet and any words that used it in their spelling. The linked blog states it was the 27th letter but gives no examples of word usage.