question about functions.

2007-12-08 2:12 pm

I cannot do this question, please help.
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Question:
Let X and Y be sets and f: X -> Y a function.
Prove that f is an injection if and only if there exists a function g: Y -> X such that
g。f=idx

Remark: f is an injection if and only if it has left inverse.
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( the sign between g and f is a little circle, which is the same as g(f(x)) )
(idx is the identity function on X)

回答 (1)

Ivan

2007-12-09 10:24 am

✔ 最佳答案

Definition: Injection: f(x1)=f(x2) => x1=x2
1)
Assume there exists a function g: Y-> X such that gf = id
If f(x1)=f(x2), then g(f(x1))=gf(x2), so id(x1) = id(x2), so x1=x2
so f is injection
2) Assume f is an injection
we need to find a left inverse of f.
so we define g as follow:
g(y) = {x if y=f(x) for some x
.......= {c if y =/= f(x) for any x, where c is a fixed point in X
Then obviously g(f(x)) = x by definition. But we need to check that g is well defined, that is, if y=f(x1)=f(x2), g(y) should give the same value, i.e. we need to show x1=x2.
but f(x1)=f(x2) => x1=x2 by injectivity of f, so we are done.

參考: PhD Math

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