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Could someone please explain to me how isomorphisms and homomorphisms work?

For an isomorphism, I know we need to follow the following four steps:

$\ \ \ $1) define a candidate,

$\ \ \ $2) show it is $1$-$1$

$\ \ \ $3) show it is onto

$\ \ \ $4) show it is closed.

But, I don't know how to actually apply it if we are given a question like what should I be focusing on.

And the same for homomorphisms.

Any help with examples would be appreciated. Thank you.

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    What do you mean by "closed"?2012-08-15
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    Also, what do you mean by "question like that"? Like what, exactly?2012-08-15
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    like phi(a)*phi(b) = phi(ab)2012-08-15
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    like just any random example that would help me understand how this isomorphism actually works2012-08-15
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    it's good to imagine isomorphism as "translation" from one language to another. For example "소고기" and "beef" look different, but they mean the same thing in Korean and English respectively.2012-08-15

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