# Thread: Problems with sets

1. ## Problems with sets

Hi, does someone know the solution to the problem:

So sorry if it has nothing to do with sets, I just have no idea what this is in English.

2. ## Re: Problems with sets

And another one, that has definitely to do with sets:

if A ⊆ B, prove that A ∪ B = B, and A ∩ B = A.

3. ## Re: Problems with sets

Have you calculated the first few $A_i$? The first problem only requires the first four and you should be able to see what others are like.
With i= 1, $A_i$ is the set of all x such that $1- \frac{1}{1}< x\le 1$ which is the same as $0< x \le 1$. In interval notation that is (0, 1].
With i= 2, $A_2$ is the set of all x so that $1- \frac{1}{2}< x\le 2$. In interval notation, that is (1/2, 2].

With i= 3, $A_3$ is the set of all x such that $1- \frac{1}{3}< x\le 3$. In interval notation, that is (2/3, 3].

With i= 4, $A_4$ is the set of all x such that $1- \frac{1}{4}< x\le 4$. 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 $X\subset Y$, then prove $Y\subset X$. And to prove that $X\subset Y$, start "if $x\in X$ and use the definitions and properties to conclude "therefore $x\in Y$".

For example, given that $A\subset B$, to show that $A\cup B= B$:
First show [b]B\subset A\cup B[/tex]. That's easy. If $x\in B$ then, by definition of "union", $x\in A\cup B$.

Now show [b]A\cup B\subset B[/tex]. If $x\in A\cup B$ then either (1) $x\in A$ or (2) $x]in B$.

1) if $x\in A$ then since we are given that $A\subset B$, $x\in B$.
2) if $x\in B$, we are done.

4. ## Re: Problems with sets

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!