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- Dictionaryquotient/ˈkwəʊʃnt/
noun
- 1. a result obtained by dividing one quantity by another.
- 2. a degree or amount of a specified quality or characteristic: "the increase in Washington's cynicism quotient"
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Jul 6, 2016 · You DO need an equivalence relation to build a quotient set, which is why the notation is S/~, which is read as "the quotient set of the set S under the equivalence relation ~." At the risk of over-simplifying it, you could say that the quotient set under a particular equivalence relation is the same as the original set, but in partitions rather than all together.
19. That operation on cosets is well-defined if and only if H is a normal subgroup. If H is just a subgroup, what you call "left quotient group" has the more standard name "set with a left group action". More precisely, the coset spaces G/H describe essentially all the examples of sets with transitive left G-actions.
Jul 18, 2018 · The axiom of choice is not needed here (assuming you can define your function $\psi:X\to Y$ without it and the resulting function $\phi$ is well-defined).
Oct 12, 2016 · Quotient ring multiplication is saying that if you multiply any two elements of I + a I + a and I + b I + b, you get an element of I + ab I + a b, which is why the quotient map is well-defined and a homomorphism. It need not be possible to write every element of I + ab I + a b as the product of two elements in I + a I + a and I + b I + b, in ...
Jul 16, 2020 · The case of a norm will follow in part from this. Suppose that d is translation invariant metric on linear topological space X compatible with the linear topology τ. Let M be a closed linear subspace in X, π the quotient map, and τM the topology on X / M induced by π. Define ρ(π(x), π(y)): = inf {d(x − y, z): x ∈ M} = d(x − y, M ...
Sep 18, 2024 · (In fact, I think one could very rarely show that even the full space G/K were open in this attempt; i.e. in general, this attempt would not even define a topology.) Much better suited might be what several commenters have implicitly or explicitly suggested:
Jun 6, 2022 · Doubt on the notation of quotient sets and quotient vector spaces 6 For a normed space, is a vector subspace of it with the restriction norm also a normed space?
T = R / Z. English: the circle group (AKA 1-torus) is the quotient group of the real numbers by the integers. The underlying homomorphism f maps each real number to their non-integer part in [0, 1), e.g. 2.34 to 0.34 and so on. Therefore, each integer e.g. 2.00 gets mapped to the origin 0 of the image as expected.
May 14, 2020 · $\begingroup$ In this old answer I try to describe the philosophy/thinking that leads to quotient groups. It's more or less the same with quotient rings: to make the algebraic structure to match the phenomenon we want to describe, we are lead to make identification of elements of some other structure.
Sep 11, 2011 · The quotient set comes with a quotient map π: X → X/R, naturally defined by sending x to its equivalence class [x]. The "quotient topology" on X/R is defined by saying " U ⊂ X/R is open iff π−1(U) is open in X." (the quotient topology itself is a special case of two things: 1) the weak topology induced by a famaily of maps too/from your ...
