Cantor's Theorem (real analysis)

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The discussion revolves around solving homework problems related to Cantor's Theorem in real analysis. For problem 1.5.6, a function f(x) was created that demonstrates a one-to-one mapping but is not onto. In problem 1.5.7, there is uncertainty about the set B, with a suggestion that it could be the empty set since no element a can be an element of f(a). The conversation emphasizes the importance of understanding these concepts rather than relying on external searches for answers. The participants are encouraged to share questions for collaborative help.
KevinL
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Google has my particular homework online. I am doing 1.5.6, 1.5.7, 1.5.8

On 1.5.6 a), I created a function f(x) such that {a} if x = a, {b} if x = b, {c} if x=c. This is 1-1 since each element of A gets mapped to something different. Its obviously not onto.

Skipping down to 1.5.7, I need to create the set B for this function. For any a, it can't be an element of f(a). Does this mean B = the empty set?

If so, I think I can tackle the rest...
 
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If you want help, then I suggest you post your question here. We're not going to do an in depth google search to find your homework.
 
In this case B is going to be the empty set
 

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