Have you calculated the first few ? The first problem only requires the first four and you should be able to see what others are like.
With i= 1, is the set of all x such that which is the same as . In interval notation that is (0, 1].
With i= 2, is the set of all x so that . In interval notation, that is (1/2, 2].
With i= 3, is the set of all x such that . In interval notation, that is (2/3, 3].
With i= 4, is the set of all x such that . In interval notation, that is (3/4, 4].
The union of those is the interval (0, 4].
For the last, to prove that two sets, X and Y, are equal, first prove , then prove . And to prove that , start "if and use the definitions and properties to conclude "therefore ".
For example, given that , to show that :
First show [b]B\subset A\cup B[/tex]. That's easy. If then, by definition of "union", .
Now show [b]A\cup B\subset B[/tex]. If then either (1) or (2) .
1) if then since we are given that , .
2) if , we are done.
That makes a lot of sense, thank you so much! I've yet to have a course in discrete math, but ran across these problems so I had no idea where to begin, but was very curious to see the solution. Now I see how obvious it was. Thanks again!