In one of the books I came across such a beautiful designation of a generalized Cartesian product. I tried to repeat it for a long time and nothing happened.
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How can i write my own math operator with limits? I want it to look like:  \sum\limits_{e=1}^{m} but with a capital A (if possible bigger than the normal text) instead of the sum.  Thanks for the help!
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Actually I have no idea how to begin. I tried to represent it as a product of two series or as a sum of two series but have no result with this. Any idea ?
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$\renewcommand{\phi}{\varphi}$ $\require{AMScd}$ My question is different from .  I want to see how in a preadditive category $\mathcal{C}$ a product is a coproduct, following . I'll write the proof up to the point where I get stuck.  Let $A=A_1,B=A_2$ be objects and suppose their product $P$ does exist, with projections $\pi_i:P\rightarrow A_i$. By considering a wedge  $$\begin{CD}A @&lt;{id_A}&lt;&lt;  A @&gt;{0_{AB}}&gt;&gt;  B\end{CD}$$  and the obvious analogous one with $A$ and $B$ exchanged, where $0_{A,B}$ is the zero morphism, we see that the product property implies the existence of two unique morphisms $k_i\in\mathcal{C}[A_i,P]$ s.t.  \begin{align*} \pi_j\circ k_i=\begin{cases} \mathrm{id}_i &amp;\text{if }i=j\\ 0_{A_i, A_j} &amp;\text{if }i\neq j \end{cases}. \end{align*} Using these equations and the definition of zero morphism, it's easy to see that the morphism $m\equiv k_1\circ \pi_1+ k_2\circ \pi_2$ satisfies $\pi_i\circ m=\pi_i$.   This is where I get stuck: In the linked proof, they now state that this implies $m=\mathrm{id}_P.$ But I don't see why this should be true.   Specifically, if $\phi\in\mathcal{C}[S,P]$ for some arbitrary object $S$, it is not apparent why $m\circ\phi=\phi$. All that we know is that there are some morphisms $x_i\in\mathcal{C}[S,A_i]$ that factor through $P$ via the universal property, but these are not necessarily unique.  And we learn even less about $m$ by looking at morphisms in $\mathcal{C}[P,S]$.  What am I missing here?
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From Rosen's Discrete Mathematics and Its Applications, 3ed, chapter 8.1 p. 506-507:    ...there are Ck ways to insert parentheses in the product x0 · x1 · · · · · xk  I would like to ask where this conclusion comes from.  I am at a loss here.  It feels that this proof does not give me the whole story.  Thank you.
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Can anyone think of an intuitive explanation of the product rule? I'm not asking for a derivation. To me it seems like quite an un-untuitive result, as apposed to the chain-rule (which is ironically harder to derive).
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Is the finite Cartesian Product of Countable sets countable?  If possible, could you give a bijection that would work for any example?  Any help would be appreciated.
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Can someone recommend a good basic book on Geometry?  Let me be more specific on what I am looking for.  I'd like a book that starts with Euclid's definitions and postulates and goes on from there to prove thereoms about triangles, circles and other plane shapes.  I'm not interested (at this time) in a book that ties geometry in with linear algebra or explains non-euclidean geometry.  All the books I've seen seem to be of two types 1) Let us use linear algebra and other techniques to put geometry on a firm footing. Let us assume you already know Eucliean geometry and move on to more interesting stuff. These are probably too advanced for me. 2) Let us skip all that complicated postulates/proofs and make geometry fun.  Too easy.  When I was in 9th grade Honors Geometry way back in 1981/82 we had a textbook that, as I recall through my foggy memory, was very good and did systematically build up geometry from the propositions (5 of them?).  And it certainly didn't attempt to address Hilbert's contributions to geometry or explain non-Euclidean geometry. Unfortunately, I don't recall the title or author.  I suspect the book was used nationally though and for quite a number of years, so perhaps someone with a better memory than me can provide a title/author.  I do recall it had all the postulates and most major thereoms listed at the end of the book.  My motivation is two fold... 1.  Review the material myself in preparation for my advanced texts. 2.  Help my children in a few years when they take geometry.    Books I could get for the iPad/Kindle or free books (Google?) would be most appreciatied, but I'm not opposed to killing a tree either :) Thanks for all the help. Dave
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This is a problem from Discrete Mathematics and its Applications:   Explain why $A \times B \times C$ and $(A \times B) \times C$ are not the same.   I understand the process behind the Cartesian product of two sets:  Let $A$ and $B$ be sets. The Cartesian Product of $A$ and $B$, denoted by $A \times B$, is the set of all ordered pairs $(a,b)$, where $a \in A$ and $b \in B$. Hence, $$A \times B = \{(a,b)\mid a \in A \land b \in B\}.$$  Youre forming a set of all the possible ordered pairs where one element is a member of the first set and the other element is a member of the other set. How would you start this proof or explanation? To me these should be the same because it involves the same operation (Cartesian cross product). Usually when there are three expressions that are applied the same operation, you just go left to right, say 4 * 5 * 6. I don't understand how you would explain A x B x C is not equal to (A x B) x C. To me, using my last example, that's like explaining 4 * 5 * 6 is not equal to (4 * 5) *6.
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My professor was lecturing today and he made this statement which I was unable to verify.  (I worded it nicer)  There is no map which is both smooth and onto from $S^1$ to $S^1$$\times$ $S^1$.  When he said this he included the original "one can clearly see" statement which is usually when the most people seem to be confused in his class.  Thus, I was hesitant to ask why.  So, please no one lecture me on the fact that I should have asked anyway.  If it is really that obvious, I would rather look stupid here than in class :)
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It seems like this should be true, but I can't find the right argument.  Thanks.  Edit: How about   1) Take two disjoint closed sets in a partial product.  2) Extend them trivially to the entire product.  3) Separate them with disjoint open sets in the big product.  4) Restrict these open sets to the subproduct.  ???
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This was  on Quora. I thought about it a little bit but didn't make much progress beyond some obvious upper and lower bounds. The answer probably depends on AC and perhaps also GCH or other axioms. A quick search also failed to provide answers.
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This seems to be a common result. I've been trying to follow the bijective proof of it, which can be found easily online, but the explanations go over my head. It would be wonderful if you could give me an understandable explanation of the proof and let me know how I'd go about finding such a bijection.
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As in the title, I was wondering if the formula: $$a\times (b\times c)=b(a\cdot c)-c(a \cdot b)$$ for $\mathbb R ^3$ cross product has some geometrical interpretation. I've recently seen a proof (from Vector Analysis - J.W. Gibbs) that's not at all difficult to understand, however I would hardly remember the steps of the proof (and I keep forgetting the correct order of A,B,C's), since it appears to me just as algebraic manipulation. So, why is this true? Or is it just an accident?
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Hi I'm having trouble understanding why the quaternionic group cannot be written as a semi-direct product in a non-trivial way. Is there a proof of why this is the case?
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Let $(X, \mathcal{J})$ and $(Y, \mathcal{F})$ be two measure spaces. Let us assume that $J$ is a collection of subsets of $X$ which generates $\mathcal{J}$, i.e. $\sigma(J)=\mathcal{J}$. Similarly, assume $\sigma(F)=\mathcal{F}$. Is it always true that $$ \sigma(J)\times\sigma(F)=\sigma(J\times F)? $$ Here, $J\times F$ is the set of all cartesian products of sets in $J$ with sets in $F$.   Please note: in the above expression the two product signs mean different things. On the left I am considering the product sigma algebra $\mathcal{J}\times \mathcal{F}$, while on the right I am considering the $\sigma$-algebra generated by the cartesian products of my ``elementary sets.''
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There is a lesser known generalization of Euler's Theorem. Here 'x' and 'm' are NOT COPRIME and $\,n\ge \log_2 m.\,$ I stumbled upon this on this site :   The "derivation" can be found towards the end of the above link; but i can't seem to wrap my head around it. It would be really great if it could be explained a bit better, probably with an example.
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Hi my course specifically talks about :  Cartesian and Polar Coordinates in 3 Dim, second Degree eqns in 3 vars, reduction to canonical forms, straight lines, shortest distance between 2 skew lines, Plane, sphere, cone, cylinder, paraboloid, ellipsoid,hyperboloid of one and two sheets and their properties.  This is merely indicative, and I would really like if the book talks about these (atleast) and some more. Would appreciate if it contained solved examples and exercise problems as well  Help Much Appreciated, Soham