Definitions o contable and uncountable numbers.\nThe gap betweeen -ve and +ve infinity.\nDefine an inverse in matematical term\nDefine a subset.\nWhat is Tensors and its theory.
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It would be countable and uncountable *sets*, not numbers.\n\nWe say that a set is countable if it is in one-to-one correspondence with some subset of the set of natural numbers (1,2,3,...). The collection of rational numbers turns out to be countable.\n\nWe say that a set is uncountable if it is not countable. ;) The set of real numbers turns out to be uncountbale, but there are many, many sets that are even larger.\n\nWe say that a set A is a subset of the set B if every element of A is in B, but not necessarily the other way around. So the set of natural numbers form a subset of the set of real numbers.\n\nA tensor is a multilinear map on a vector space. Way too much theory for this space.\n\nGiven a *function* f, an inverse function g, is one where the compositions fog and gof are both the identity map.\n\nThe gap between +ve and -ve infinity makes no real sense unless you simple mean the set of real numbers.
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Irreduicible basically means that you can't factor it.  This particular one can't be expressed as the product of two or more irreduicible polynomials.
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Minkowski space: the three ordinary dimensions of space are combined with a single dimension of time to form a four-dimensional manifold for representing a spacetime.  Formally, Minkowski space is a four-dimensional real vector space equipped with a nondegenerate, symmetric bilinear form with signature (-,+,+,+).  This signature is often called the Landau-Lifshitz (spacelike) sign convention
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~ you need to say what are V and W??\nI am guessing that V and W are vector spaces and that T is  LINEAR transformation from V --> W.\n\nsuppose  dim(V) < dim(W), take a basis of W: w1, ... ,wn, and assume that T is onto, then for every wi there exist a vi such that\nT(v1)=w1, ... , T(vn)=wn,\nnow let us check if v1,... vn are linearly independent:\ntake a1v1 + ... + anvn =0 a linear combination equal to cero, then\nT ( a1v1 + ... + anvn  ) = T(0) =0,\non the other hand,\nT ( a1v1 + ... + anvn  )  = a1T(v1) + ...+ anT(vn)\n= a1w1 + ... + anwn =0, since w1,..., wn are a basis, that means that\na1=a2=..=an=0\nbut this implies that v1, ... vn are linearly independent.\nand this is a contradiction to the fact that dim V < dim W\n\n\nfor the other problem:\nif dim(V) > dim(W), prove that T cannot be one to one (injective).\n\nAssume that v1,..., vm form a basis for V,\nif Ker(T) =0 (i.e. if we assume that T is 1-1), then\nT(v1), ... , T(vm) are a linearly independent set of vectors, but\nm=dim V > dim (W), but this cannot happen, since any set of linearly independent vectors in W would have to have fewer than \ndim(W) vectors.
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Let's start with an example of a vector space. An example is a plane. If we take any two points (we can make them vectors by drawing the points through the origin) in the plane and add them, the result will still be in the plane. If we take any scalar multiple of a vector in the plane, the result will still be in the plane. In fact, the resulting vector will be parallel to the original vector. \n\nIt's these two properties that make a vector space a vector space. In fact, what I said about the plane holds for any n-dimensional (n is any whole number that can be imagined) space. Further generalizations can be made to other examples that satisfy the addition and scalar multiplication properties mentioned above. But staying in the same space when adding or multiplying by a scalar are the key properties of a vector space.
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Inf. is not a number, so you can't directly define operations with it. You can only talk about limits, such as lim(n->inf)an*bn, where lim(n->inf)an=A and lim(n->inf)bn=B.\n\nIf A and B are finite numbers, then the limit is always A*B (with the particular case A=0 leading to A*B=0).\n\nHowever, when B=inf then:\n- if A>0 then the limit is inf ("A*inf=inf")\n- if A<0 then the limit is -inf ("A*inf=-inf")\nBUT\n- if A=0 then the limit may or may not exist, and if it exists then it can be ANY number, or inf, or -inf\n\nExamples:\n- an=(-1)^n/n, bn=n => an*bn=(-1)^n, which doesn't have a limit\n- an=x/n, bn=n (x being any number) => an*bn=x, so the limit is x\n- an=1/n, bn=n^2 => an*bn=n so the limit is inf\n- an=-1/n, bn=n^2 => an*bn=-n so the limit is -inf\n\nIn each example, lim(n->inf)an=0 and lim(n->inf)bn=inf\nTherefore, 0*inf (or inf*0) can NOT be defined.
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a physical quantity which has different magnitudes in different directions is called a tensor quantity no torque is a vector quantity whereas vector is a physical quantity which has both magnitude and direction and also obeys the laws of vector addition .example of a torque quantity is moment of inertia
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There is no problem with having an element of a set that is itself a set. Element does not imply anything about being indivisible; it simply means it satisfies the property defining the set. The set of all natural numbers is a set, and the natural numbers can themselves be defined as equivalence classes of sets having certain cardinalities. See the definition of set in the source below.\nI think you should go over your definitions. It is trivial that your set has countable cardinality, and it is not an example of any paradox.
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Although our computing tools (such as our brain) are finite, but they can touch infinity through mathematical induction or recursion.\n\nFeeling finit ocuures for a neural network, when the data stored in it is going to overfow the nerons capacity.
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Infinity.Any number divided by 0 is infinity.
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It has dimension 2.\nI will not bother to prove that this subset is indeed a subspace except to note that (0,0,0) is an element.\nThis is most readily seen by constructing a basis.\nI propose that {(1,1,0),(0,1,1)} is a basis for this space.\nLet (a,b,c) be an element of this space. Then (a,b,c)=(a,a+c,c) =  a*(1,1,0) + c*(0,1,1)\nThis our proposed basis Spans this subspace.\nNow I think we can clearly see that our proposed basis is linearly independant since:\nIf:   a*(1,1,0) + c*(0,1,1) = (0,0,0)\nthen: 0 = a, 0 = a+c and 0 = c by equating the components.\nThus we have a linearly independant spanning set.\nThus we have a basis!\nSince our basis is of magnitude 2, our space must be of finite dimension 2.
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There won't be a book specifically for tensor analysis. Depending on your background, you're most likely to find what you want in either a General Relativity book or a differential geometry book. For a non-axiomatic approach, you probably want the Relativity book. Warner's book is quite good.\n\nOne difficulty with the subject is that mathematicians and physicists do things very differently when it comes to tensors. In physics, a coordinate system is often assumed and the thing that defines a tensor is the transformation properties between coordinate systems. For mathematicians, the crucial thing is multi-linearity on some vector space. Because of this, there is a *lot* of confusion when math and physics people talk about tensors. GOOD LUCK!
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Mathematically n dimensions can be described but our experience is only 3 dimensions So the properties of fourth dimensonal shapes can be mathematically described and a fourth dimensional quantity that is in units^4 can also be calculated\n\nThis is an argument that the 4th dimension is time. That cannot be the case when one considers the properties in 3d are L, L, and L One would immediately envisage that the 4th, 5th .... nth dimensions would also all be L\n\nA tesseract, the 4 dimensional equivalent to the 2 dimensional square and 3 dimensional cube, is not a time related object at all. Its 3D projection can be build and the objects equivalent of a 2D shadow seen easily. In 4-space it occupies L^4 units^4. Wikepedia has some interesting information and projections of a tesseract
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okay so what N means is that you are plugging in any integer value for n. integers, as a reminder, a whole, countable numbers like 1, 6, 88, 45798.\nSo teh first four integers (values of n) are obvious 1, 2, 3, 4\nso the sequence is\n(2*1 -5), (2 *2 -5), (2 *3-5), (2 *4-5)\n-3, -1, 1, 3.\n\nHere's one way to prove that it's arithmetic:\nYou can graph them as continuous variables along an x y coordinate system. That is, graph the points (1, -3), (2, -1), (3, 1), and (4, 3). If you connect them, you'll get a straight line. It's no surprise that the equation of that line (if you choose to find it) is y=2x-5. Because the points lie on a straight line, it's arithmetic (as opposed to geometric).
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Hint:\nPick y and z different. Let \nf(x)=d(x,y)-d(x,z).\nThen f(y)<0 and f(z)>0. Show that the range of f is uncountable.
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easy infinity +1\n\nThe limit that a function is said to approach at x = a when (x) is larger than any preassigned number for all x sufficiently near a. \n\nThe key word here is preassigned.  In this case there can never be a true infinity.  You have to define the set that you are taking to infinity.  Whether it be numbers, distances, time, space, etc.  Just by defining the set, puts a limit on your answer, which makes that limit infinity.  And any comparable limit is infinity as well, unless it's exactly the same as the first.
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ehh its more a math term. in terms of math a conjugate would be something like if you had the given as: 3x-4, the conjugate would be: 3x+4. in essence it is the opposite as the original set of numbers. in conversation im not totally sure how you would use that and someone else may be of more knowledge there.. but ive only heard it in respects to math.