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A question concerning problem 4 in HW1 and a general question about proofs

A question concerning problem 4 in HW1 and a general question about proofs

av Shamiur Rahman Ramim -
Antal svar: 4

Hi! I have some questions regarding problem 4 and one about proofs in general

In the homework the union $\bigcap_{\alpha} A_{\alpha}$ is displayed. Is $\alpha$ an element of some index set?

Does it matter whether or not this index set is at most countable or uncountable? 

Could I in the solution write $\bigcap_{\alpha \in L} A_{\alpha}$, where L is an index set?

How much do I need to write in the solutions for it to be OK? If I define a function, that is *clearly* a bijection (for example f : {1, 2} -> {3, 4}, with f(1) = 3, f(2) = 4), do I need to prove that it is a bijection? And do I need to prove that every function I define is well-defined? 

Best regards, Shamiur

Som svar till Shamiur Rahman Ramim

Re: A question concerning problem 4 in HW1 and a general question about proofs

av Sofia Tirabassi -

Hello,

yes $\alpha$ is in some index set and if no assumption is given it actually can be uncountable. You can certainly write it more expliccitly in your solution as you suggest.


What you propose is clearly a bijection so no further explanation is needed :)

Well defined: you do not need to prove it every time, but just when it is necessary. This might seems circular but is not. If the function law obviousy gives just one output for every input you do not need to prove well definedness. On the other side, if you have a function from a set of set using elements to decide the value, you have to make sure that the function does not change if you change the element. In case like this well defined has to be proven.

Som svar till Sofia Tirabassi

Sv: Re: A question concerning problem 4 in HW1 and a general question about proofs

av Shamiur Rahman Ramim -

Hello!

I'm not sure if I'm really understanding what you mean by 

"On the other side, if you have a function from a set of set using elements to decide the value, you have to make sure that the function does not change if you change the element".

If I'm understanding correctly, wouldn't the example I gave (f : {1,2} -> {3,4}, f(1) = 2, f(3) = 4) be a function I have to prove well-definedness for? 

If I'm not understanding correctly, could you please elaborate?

Best regards, Shamiur.

Som svar till Shamiur Rahman Ramim

Re: Sv: Re: A question concerning problem 4 in HW1 and a general question about proofs

av Sofia Tirabassi -
Hello agai,
sorry I was not clear. The function you porvide does not need showing that is well defined, since clearly every element on the left is associated to an element on the right.
You need to show well definednes when you have a functions from a set whose element are not uniquely defined, like a quotient set. If the law of the function depends from the definition of the elements, then, in that case you need to show that your function is well defined. Not in the example that you propose.