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Question Number 126808 by sdfg last updated on 24/Dec/20

Answered by JMZ last updated on 24/Dec/20

Since φ is a homomorphism  φ(a+b)=φ(a)+φ(b).  The kernel contais every x which

$${Since}\:\phi\:{is}\:{a}\:{homomorphism} \\ $$$$\phi\left({a}+{b}\right)=\phi\left({a}\right)+\phi\left({b}\right). \\ $$$${The}\:{kernel}\:{contais}\:{every}\:{x}\:{which} \\ $$

Answered by mindispower last updated on 24/Dec/20

⇔ϕ(a−23)=0  ⇒a−23∈Ker(ϕ)  ⇒a−23∈{0,10,20}  a∈{23,3,13}

$$\Leftrightarrow\varphi\left({a}−\mathrm{23}\right)=\mathrm{0} \\ $$$$\Rightarrow{a}−\mathrm{23}\in{Ker}\left(\varphi\right) \\ $$$$\Rightarrow{a}−\mathrm{23}\in\left\{\mathrm{0},\mathrm{10},\mathrm{20}\right\} \\ $$$${a}\in\left\{\mathrm{23},\mathrm{3},\mathrm{13}\right\} \\ $$

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