I'm presented with this,(adsbygoogle = window.adsbygoogle || []).push({});

[tex](1)\ (\bigcup^{\infty}_{\n=1}A_{n})^{c}\ =\ \bigcap^{\infty}_{\n=1}A_{n}^{c} [/tex]

and asked why induction cannot be used to conclude this.

Now, i know the principle behind induction is to show that P(S)=N by showing that when

(i) S contains 1 and

(ii) whenever S contains a natural number n, it also contains n+1.

So I assume somewhere along the line (1) cant fit into either (i) or (ii), but i'm just not getting it. Help please, thanks.

I also have some trouble understanding infinity in general. My text says that, notationally,

[tex](2)\ \bigcup^{\infty}_{\n=1}A_{n}[/tex]

is just another way of saying the set whose elements consist of any element that appears in at least one particular (An), but, in a previous question, we already proved with induction that

[tex](3)\ (A_1\ \cup\ A_2\ \cup\ A_3\ \cup\ ...\ \cup\ A_n)^c\ =\ A^{c}_{1}\ \cup\ A^{c}_{2}\ \cup\ A^{c}_{3}\ \cup\ ...\ \cup\ A^{c}_{n}[/tex]

In my mind, (3) should lead to (1), and since we used induction on (3) to extrapolate to (1), then we did use induction to prove (1), but obviously, this is incorrect and im not sure why?

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# Why can i not use induction here?

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