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Consider the map φ: Z × Z → Z via (a, b) → a - b between the groups Z × Z and Z (both under addition). (a) Show φ is an epimorphism. (b) Determine K = ker φ. (c) Describe the set φ^(-1)(5) (i.e. all elements that map to 5). (d) State the isomorphism obtained by applying the First Homomorphism Theorem to φ.
Submitted by Carlos E. Aug. 17, 2021 06:15 a.m.
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10 comments
Krista R.
April 4, 2023
This was super helpful, seriously can't thank ya enough!
Molly H.
May 26, 2023
Youve got a real talent for breakin' down tricky stuff in an easy way.
Melissa P.
June 4, 2023
This vid totally opened my eyes, thanks a mill
Melanie M.
July 8, 2023
Well, Mr S is breakin it down for us on how to show that phi is an epimorphism Seems like hes got the math skills to prove that phi is a homomorphism first and then onto But, whats the deal with the pre-image of the element mapping to 5 Can we get some mo
Rita C.
August 4, 2023
mr ss explanation that phi is a homomorphism and onto is cool but im still curious about k ker phi. can someone sprinkle some more details on what k actually is and whats the isomorphism obtained using the first homomorphism theorem applied to phi
David R.
September 5, 2023
the answer gave a clear explanation of how the function meets the criteria for being an epimorphism.
Angela L.
September 16, 2023
The answer didn't include a description of the set φ^(-1)(5), which was asked in part (c) of the question.
Charles N.
September 25, 2023
The use of the term "onto and "homomorphism was well explained in the answer.
Heather J.
September 29, 2023
The explanation of the function φ: Z × Z → Z as an epimorphism was well-elaborated.
Alexis F.
September 30, 2023
The answer could have included more detailed steps for determining the ker φ in part (b) of the question.