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Consider functions f and g such that composite gof is defined and is one-one.Are f and g both necessarily one-one. |
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Answer» Let there are two functions `f` and `g` such that `f: A -> B and g: B -> C` Now `gof: A -> C` Given that `gof` is one-one. To prove that `f: A -> B` is one-one, we have to prove that if `f(x) = f(y)` then `x = y` for all `x, y in A`. Now let `x, y in A` such that `f(x) = f(y)` Then `gof(x) = g(f(x)) = g(f(y))` `=> gof(x) = gof(y)` `=> x = y` (since `gof(x)` is one-one) Since `gof` is one-one, hence it shows that `f` is one-one. Again `g` may or may not be one-one. So `gof` is one-one does not imply that both `f` and `g` has to be one-one. |
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