Isomorphisms and Homomorphisms Calculator

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Example

Created on 2024-06-20Asked by Scarlett Green (Solvelet student)

Let

f: \mathbb{R} \to \mathbb{R}

be defined by

f(x) = 2x + 3

. Determine if

f

is a homomorphism.

Solution

To determine if

f(x) = 2x + 3

is a homomorphism, we need to check if

f

preserves the operation of addition, i.e., if:

f(x + y) = f(x) + f(y)

Calculate

f(x + y)

f(x + y) = 2(x + y) + 3 = 2x + 2y + 3

Calculate

f(x) + f(y)

f(x) + f(y) = (2x + 3) + (2y + 3) = 2x + 2y + 6

Since

f(x + y) \neq f(x) + f(y)

f

is not a homomorphism. Solved on Solvelet with Basic AI Model

Some of the related questions asked by Aria Scott on Solvelet

1. Determine whether the mapping

T : \mathbb{R} \to \mathbb{R}

defined by

T(x) = 2x

is an isomorphism,2. Compute the Laplace transform of the

n

th derivative of the function

f(t) = t^n

using the differentiation property of Laplace transforms.,

DefinitionHomomorphism an Isomorphism - Through this article, we will also get a taste of abstract algebra with fancy names such as isomorphisms and homomorphisms. A homomorphism is a map that retains the operations of two algebraic structures. An isomorphism is a bijective homomorphism, and if it has an inverse, too, which is also a homomorphism, then the two structures are structurally identical. Example: A group homomorphism ϕ:G→H satisfies ϕ(ab)=ϕ(a)ϕ(b) for every a,b∈G. ϕ is bijective if it is an isomorphism, meaning that G and H are the same structurally.

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