# Math Help - MCQ Question on sets (it seems like calculus ;))

1. ## MCQ Question on sets (it seems like calculus ;))

The set

is identical to :

1. (-1 , 1)
2. [-1 , 1)
3. (-1 , 1]
4. [-1 , 1]

--------------------

The set

is identical to :

1. (-1 , 1)
2. [-1 , 1)
3. (-1 , 1]
4. [-1 , 1]

2. For n=1,2,3,4 the interval $(-1-n^{-1}, 1+n^{-1})$ is equal to $(-2,2),\;(-\tfrac32,\tfrac32),\;(-\tfrac43,\tfrac43),\;(-\tfrac54,\tfrac54)$. You can see that as n increases, all these sets are going to contain the interval [–1,1] (including the endpoints). But as n gets very large, all points to the left of –1 or to the right of +1 are eventually going to be excluded from $(-1-n^{-1}, 1+n^{-1})$. So the intersection of all these intervals is exactly [–1,1].

The other part of the question appears to ask for the union of the same sequence of intervals. The union will be the largest of the intervals, which is the first one, namely (–2,2). However, I think that you probably intended to ask for the union $\textstyle\bigcup_{n=1}^\infty(-1+n^{-1},1-n^{-1})$. I'll leave you to work out what that is. I suggest that you use the same method as for the previous part: start by writing down what happens for n=1,2,3,4,... and then think about what happens as n gets very large.

3. actually, I was interpreting [-1,1] as a set including all values from -1 to 1 . so a set something like {-1, -0.99,-.98........0.99,1}
that is why i was unsure on how the answer could be [-1,1]

Any ya, so the answer to part two is also [1,-1] right?

4. Originally Posted by champrock
Any ya, so the answer to part two is also [1,-1] right?
NO! The fact is the answer to #2 is $(-2,2)$.
But that is not given.
Did you type it correctly?

5. ya just noticed that it was a typo. i copied the same values from the first question

6. one more doubt in the second question

When I apply limit to (-1+1/n) upto infinity, i get -1. So, does that mean that -1 should be included in the interval?
Similarly, should 1 be included in the interval?

so intervals should be (-1,1) or [-1,1] ?

7. $\begin{gathered}
\bigcap\limits_{n = 1}^\infty {\left( { - 1 - n^{ - 1} , 1 + n^{ - 1} } \right)} = \left[ { - 1,1} \right] \hfill \\
\bigcup\limits_{n = 1}^\infty {\left( { - 1 - n^{ - 1} , 1 + n^{ - 1} } \right)} = \left( { - 2,2} \right) \hfill \\
\end{gathered}$