Determine if the following functions are homomorphisms / embeddings / isomorphisms. You must only answer: "not a homomorphism", "homomorphism buffnot an embedding", "embedding but not an isomorphism", "isomorphism". If not specified, the interpretations of the symbols in the signatures are the standard ones. a. n ↦ 2n from (ℕ, 0, +, ·) to (ℕ, 0, +, ·). b. n ↦ 2n from (ℕ, 0, +) to A := (ℕ, 0A, +A), where 0A is 1 and +A is the usual multiplication. c. (x, y) ↦ x - y from A := (ℝ2, 0A, +A) to (ℝ, 0, +), where 0A := (0, 0) and +A is the usual coordinate-wise addition of points in ℝ2.
Determine if the following functions are homomorphisms / embeddings / isomorphisms. You must only answer: "not a homomorphism", "homomorphism buffnot an embedding", "embedding but not an isomorphism", "isomorphism". If not specified, the interpretations of the symbols in the signatures are the standard ones. a. n ↦ 2n from (ℕ, 0, +, ·) to (ℕ, 0, +, ·). b. n ↦ 2n from (ℕ, 0, +) to A := (ℕ, 0A, +A), where 0A is 1 and +A is the usual multiplication. c. (x, y) ↦ x - y from A := (ℝ2, 0A, +A) to (ℝ, 0, +), where 0A := (0, 0) and +A is the usual coordinate-wise addition of points in ℝ2.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Determine if the following functions are homomorphisms / embeddings / isomorphisms.
You must only answer: "not a homomorphism", "homomorphism buffnot an embedding", "embedding but not an isomorphism", "isomorphism".
If not specified, the interpretations of the symbols in the signatures are the standard ones.
a. n ↦ 2n from (ℕ, 0, +, ·) to (ℕ, 0, +, ·).
b. n ↦ 2n from (ℕ, 0, +) to A := (ℕ, 0A, +A), where 0A is 1 and +A is the usual multiplication.
c. (x, y) ↦ x - y from A := (ℝ2, 0A, +A) to (ℝ, 0, +), where 0A := (0, 0) and +A is the usual coordinate-wise addition of points in ℝ2.
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