- #1
RespeckKnuckl
- 7
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Validity of proof method -- error in book?
Hey guys, I'm going through a book on category theory (not homework, not for a class) and I'm having trouble following a provided proof, I think there's something wrong with his reasoning. You don't necessarily need to know category theory to understand my question. Essentially he wants to prove that every monomophism in Set is an injective function. So he starts by proving that if f is an injective function, then it is a monomorphism (part 1). Then he tries to prove that if f is a monomorphism, it is an injective function (part 2). There is no problem with part 1, so I will just write part 2 here:
"...let f:B->C be a monomorphism. If f is not injective, then there are distinct elements b,b' in B for which f(b)=f(b'). Let A be the one-element set {a}, and let g:A->B map a to b while h:A->B maps a to b'. Then f(g(a)) = f(h(a)), contradicting the assumption that f is a monomorphism."
Something seems very wrong to me here. He wants to show that for all f: f being a monomorphism ( m(f) ) implies that f is injective ( i(f) ), and he goes about it with a proof by contradiction: it is impossible for m(f) and not i(f) to be true. But rather than showing it happens in call cases, he then defines a very specific example (the set A) and shows that in that particular case, m(f) and not i(f) lead to a contradiction. But from that you can't extrapolate that for ALL f, m(f) and not i(f) lead to contradictions. Am I missing something here? Or should I abandon this book and find a better one?
Hey guys, I'm going through a book on category theory (not homework, not for a class) and I'm having trouble following a provided proof, I think there's something wrong with his reasoning. You don't necessarily need to know category theory to understand my question. Essentially he wants to prove that every monomophism in Set is an injective function. So he starts by proving that if f is an injective function, then it is a monomorphism (part 1). Then he tries to prove that if f is a monomorphism, it is an injective function (part 2). There is no problem with part 1, so I will just write part 2 here:
"...let f:B->C be a monomorphism. If f is not injective, then there are distinct elements b,b' in B for which f(b)=f(b'). Let A be the one-element set {a}, and let g:A->B map a to b while h:A->B maps a to b'. Then f(g(a)) = f(h(a)), contradicting the assumption that f is a monomorphism."
Something seems very wrong to me here. He wants to show that for all f: f being a monomorphism ( m(f) ) implies that f is injective ( i(f) ), and he goes about it with a proof by contradiction: it is impossible for m(f) and not i(f) to be true. But rather than showing it happens in call cases, he then defines a very specific example (the set A) and shows that in that particular case, m(f) and not i(f) lead to a contradiction. But from that you can't extrapolate that for ALL f, m(f) and not i(f) lead to contradictions. Am I missing something here? Or should I abandon this book and find a better one?