1) Give an example of a map from Z to Q that is a homomorphism for which 1 in Z is neither mapped to 1 nor -1 in Q (operation in Z and Q is addition)
2) Construct a homomorphism from Z to Q that is not an injection.
2) Construct a homomorphism from Z to Q that is not an injection.
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1) Define f : Z → Q by f(n) = n/2.
Then, f(1) = 1/2, and for any a, b in Z, f(a+b) = (a+b)/2 = a/2 + b/2 = f(a) + f(b).
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2) Define g : Z → Q by g(n) = 0 for all n.
I hope this helps!
Then, f(1) = 1/2, and for any a, b in Z, f(a+b) = (a+b)/2 = a/2 + b/2 = f(a) + f(b).
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2) Define g : Z → Q by g(n) = 0 for all n.
I hope this helps!