when i was at uni in my final year the lecturer gave this definition@\nan isometric isomorphism is an isomorphism that is isometric. needless to say it didn't help.
201812
Isomorphisms between groups are not the same thing as isomorphisms between, for example, rings.  There is a fully general definition of isomorphism that stems from universal algebra and mathematical logic, but I risk boring you if I develop it here.  So let me tackle the specific case of normed vector spaces, as I suspect it is in this setting that you came across the term "isometric isomorphism".\n\nTwo normed vector spaces are said to be isomorphic if there is a bijective map f between them that is homogeneous and linear, and for which there is a constant C>0 such that (1/C) ||x|| < ||f(x)|| < C ||x|| for all x.  An isometric isomorphism is an isomorphism for which that constant C can be taken to be 1, provided you reinterpret the symbol < as less than or equal--it's not easy to type the appropriate symbol.  In such a case, distances are not only roughly preserved (as in the case in which one has an isomorphism with constant C>1): they are indeed actually exactly preserved.  Thus the name isometry.
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it means exactly what it says.\nyou failed to supply a required argument for a function/object
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A rotation of 0 degrees doesn't move or transform the figure at all.  It's like sliding it by a vector of <0, 0>, or enlarging it by a scale factor of 1.\n\nNo movement = no transformation.\n\nYour isosceles right triangle will have line symmetry, only.
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R2:\nis transitive since (a,b) in R2 and (b,c) in R2, then a<b and b<c so a<b<c and (a,c) is in R2.\nis not reflexive, since (a,a) is not in R2, because a=a and so a is not less than a.\nis not symmetric since (a,b) is in R2, thus a<b and b is not less than a, thus (b,a) is not in R2.\n\nR1:\nis transitive as above.\nis reflexive, since a=a, so a leq a, thus (a,a) in R1\nis not symmetric, since (1,2) in R1, but (2,1) is not in R1.\n\n\nR3:\nis transitive, as above.\nis reflexive, as above.\nis symmetric, since (a,b) in R3, thus a=b and b=a.  Therefore (b,a) in R3.\n\nThus R3 is the only "equivalence relations," or a relation that is reflexive, symmetric, and transitive.\n\ngood question.
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If A is a subset of C and A has the same cardinality as C, then the sets A and C are the same.\n\nI guess as a proof, you could assume that there is an element in C that is not in A...in which case the cardinality of C would be greater than the cardinality of A, which is a contradiction.\n\nThen B gets trapped in the middle.  A = B = C.
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Yes,\nbecause you cannot have two things contradicting each other yet both of them are correct. for example, a>= b and a<b , they cannot be both correct.\n\nWhich one is correct then? well that's for you to find out.\n\nRegards
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An object is symmetrical if it could be cut down the middle and produce two mirror images.  "Bilateral" just means "two sides", indicating that the object is symetrical in one direction.  The term is used in biology, and is a common trait of the animal kingdom.  \n\nFor instance, look at yourself.  You have one eye on either side of your nose, one ear on either side of your head, one arm on either side of your shoulders and one leg on either side of your hips.  If you took a picture of yourself and cut it down the middle, from top to bottom, the two pictures would be mirror images of one another.  This is bilateral symmetry, the same on both sides.
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i think its true\nim f is isomorphical  with G/(ker f) so\n|G| = |ker f|*|G/(ker f )| = |ker f| *|im f|.\n i applied two well known theorems.
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Bob, I used to enjoy your show before you started doing all those really expensive houses.  Then, of course, you left.\n\nOh, well.\n\nDon't you know, Bob, that isosceles triangles don't have "right sides?"  An isosceles triangle has two "legs" of equal length.  If it is a right isosceles triangle, the side opposite the right angle is called the hypotenuse.\n\nSay hi to Norm for me!
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Range: x>0\nSymmetry: About the origin, x = 0\nIntervals: Increasing: x < 0\n               Decreasing: x > 0\nLocal min: x = 0\nLocal max: none, diverges as x => (goes to) 0\n\n  1/0 is undefined, so y approaches infinity as x gets smaller and smaller.
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It's known as the isosceles three way: /_&#92;
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2-M 2-A 2-T Therefore the term in the denominator. Wish you could've posted the exact question. I think -1 has been used to mean rearrangement ie. In what other ways in which MATHEMATICS can be rearranged, leaving alone this particular arrangement MATHEMATICS. If you are a JEE (IIT) aspirant I'd like to meet you. Mail me.
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just a piont of view, but normally to divide something symmetrically you have to do it at the widest piont along the centre line. where as assymetricaly is off the centre line at a narrower piont and therefore not a symmetry,this only applies to poloygons and symmetric shapes.  hope this helps
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Additive equivalence is a symmetric relation over the reals. See the source link.
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An Object is exactly what its meant to mean, it is an object. In terms of OO Programming, an object is an instance of a class and methods are functions that define a class actions when invoked. \n\nFor example: \n\nYou create a class Automobile.\nA Car is defined as an object of the class Automobile because a car is an automobile.\n\nAn automobile will have a number of miles, so you create a method in class called getMiles(). \n\nIn a class you declare a set of variables for a possible outcome. \n\nFor more detailed and clear information on OO progrramming, log on to www.sun.com.
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Yes, \n1:1 means 1 and by convention, \nthis means 1 to 1 matching i.e.same \nimplying  'same dimensions as the other'\nSimple...Got it?
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The first thing is to understand the difference between similar and congruent (identical).  Two triangles are similar if their shapes are similar, but their sizes may be different.  An equilateral triangle is similar to another equilateral triangle, but one could have sides of length 1 and the other sides of length 2.  These are *similar*.  But for them to be congruent both their angles and their sides would all have to be the same.\n\nLike you said, if two triangles had 2 angles the same (and therefore the 3rd angle must be known to be the same since they all add up to 180), then you've only proven them *similar*.  A side could be 3 units long, but if you doubled them all, you would still have a *similar* triangle, but it wouldn't be *congruent*.\n\nA concept commonly taught in high school mathematics is that of proving the "angle" and "side" theorems, which can be used to define two triangles as similar or congruent.\n\nIn each of these three-letter acronyms, A stands for equal angles, and S for equal sides. For example, ASA refers to an angle, side and angle that are all equal and adjacent, in that order.\n\nAAA - Angle-Angle-Angle. If two triangles share three common angles, they are *similar*. (Obviously, this means that the side lengths are locked in a common ratio, but can vary proportionally, making the triangles similar.) Additionally, since the interior angles of a triangle have a sum of 180°, having two triangles with only two common angles (sometimes known as AA) implies similarity as well. \n\nTwo triangles are *congruent* if their corresponding sides and angles are equal. Usually it is sufficient to establish the equality of three corresponding parts and use one of the following results to conclude the congruence of the two triangles:\n\nSAS (Side-Angle-Side): Two triangles are *congruent* if a pair of corresponding sides and the included angle are equal.\n\nSSS (Side-Side-Side): Two triangles are *congruent* if their corresponding sides are equal.\n\nASA (Angle-Side-Angle): Two triangles are *congruent* if a pair of corresponding angles and the included side are equal.\n\nWhile the AAS (Angle-Angle-Side) condition also guarantees *congruence*, SSA (Side-Side-Angle) does not, as there are often two dissimilar triangles with a pair of corresponding sides and a non-included angle equal. This is known as the ambiguous case. Of course, AAA (Angle-Angle-Angle) says nothing about the size of the two triangles and hence shows only *similarity* and not congruence.\n\nHowever, a special case of the SSA condition is the HL (Hypotenuse-Leg) condition. This is true because all right triangles (which this condition is used with) have a congruent angle (the right angle). If the hypotenuse and a certain leg of a triangle are congruent to the corresponding hypotenuse and leg of a different triangle, the two triangles are congruent.