prob and stat inference-sets review

kid funky fried · #1 $Old$ August 20th, 2008, 08:36 PM

I need to review sets for my probability and statistical inference course.
I do not understand the meaning of the following:

$\begin{gathered} A_k = \left\{ {x:\frac{{10}}{{k + 1}} \leqslant x \leqslant 10} \right\},k = 1,2,3,.... \hfill \\ \hfill \\ \bigcap\limits_{k = 1}^8 {A_k } = \left\{ {x:5 \leqslant x \leqslant 10} \right\} = A_1 \hfill \\ \end{gathered}$

mr fantastic · #2 $Old$ August 21st, 2008, 12:44 AM

Quote:

Originally Posted by kid funky fried $View Post$

I need to review sets for my probability and statistical inference course.
I do not understand the meaning of the following:

$\begin{gathered} A_k = \left\{ {x:\frac{{10}} {{k + 1}} \leqslant x \leqslant 10} \right\},k = 1,2,3,.... \hfill \\ \hfill \\ \bigcap\limits_{k = 1}^8 {A_k } = \left\{ {x:5 \leqslant x \leqslant 10} \right\} = A_1 \hfill \\ \end{gathered}$

The first statement defines the set $A_{k}$ for k = 1, 2, 3, .....

$A_1 = \{ x: \,5 \leq x \leq 10 \}$

$A_2 = \{ x: \,10/3 \leq x \leq 10 \}$

$A_3 = \{ x: \,5/2 \leq x \leq 10 \}$

etc.

$A_8 = \{ x: \,5/4 \leq x \leq 10 \}$

The second statement says that the intersection of the eight sets $A_1$ , $A_2$ , $A_3$ , ....... $A_8$ is equal to $\{ x: \,5 \leq x \leq 10 \}$ (which is true by the way), which of course is just the set $A_1$ .

kid funky fried · #3 $Old$ August 21st, 2008, 05:52 PM

I think I understand now.
The intersection is equal to A1 because A1 is the only set that would be common in all the other sets ( A2 through A8).

Thanks Again.

Kid