Answer
(a) \((0, \infty)\)
(b) \((4, \infty)\)
(c) Not mutually disjoint
(d) \((0, \infty)\)
(e) \((n, \infty)\)
(f) \((0, \infty)\)
(g) \(\varnothing\)
Work Step by Step
We're given:
\[
W_i = \{ x \in \mathbb{R} \mid x > i \} = (i, \infty) \quad \text{for all nonnegative integers } i.
\]
We'll compute the unions and intersections of these open intervals.
---
## **(a)** \(\displaystyle \bigcup_{i=0}^4 W_i = \bigcup_{i=0}^4 (i,\infty)\)
Notice:
- \(W_0 = (0, \infty)\)
- \(W_1 = (1, \infty)\)
- \(W_2 = (2, \infty)\), etc.
The **union** will be the largest of these, i.e., the one that starts earliest. Since all sets are subsets of \(W_0\), we get:
\[
\boxed{\bigcup_{i=0}^4 W_i = (0, \infty)}
\]
---
## **(b)** \(\displaystyle \bigcap_{i=0}^4 W_i = \bigcap_{i=0}^4 (i,\infty)\)
Now we’re taking the intersection — only values that are in **all** of the intervals.
- The intersection of \((0,\infty), (1,\infty), (2,\infty), (3,\infty), (4,\infty)\) is the smallest among them:
\[
\bigcap_{i=0}^4 W_i = (4, \infty)
\]
Hence:
\[
\boxed{\bigcap_{i=0}^4 W_i = (4, \infty)}
\]
---
## **(c)** Are \(W_0, W_1, W_2, \dots\) mutually disjoint?
No.
For example:
- \(x = 5\) is in **every** \(W_i\) where \(i < 5\).
- That means the sets overlap.
So the sets are **not** mutually disjoint.
\[
\boxed{\text{No, they are not mutually disjoint.}}
\]
---
## **(d)** \(\displaystyle \bigcup_{i=0}^n W_i = \bigcup_{i=0}^n (i,\infty)\)
The smallest interval in this union is \(W_0 = (0, \infty)\), which contains all the rest. So:
\[
\boxed{\bigcup_{i=0}^n W_i = (0, \infty)}
\]
---
## **(e)** \(\displaystyle \bigcap_{i=0}^n W_i = \bigcap_{i=0}^n (i,\infty)\)
The intersection is the most restrictive (largest lower bound). The largest \(i\) is \(n\), so:
\[
\boxed{\bigcap_{i=0}^n W_i = (n, \infty)}
\]
---
## **(f)** \(\displaystyle \bigcup_{i=0}^{\infty} W_i = \bigcup_{i=0}^\infty (i, \infty)\)
The earliest interval is \(W_0 = (0,\infty)\), and since all others are subsets of it, the union stays:
\[
\boxed{\bigcup_{i=0}^\infty W_i = (0, \infty)}
\]
---
## **(g)** \(\displaystyle \bigcap_{i=0}^{\infty} W_i = \bigcap_{i=0}^\infty (i, \infty)\)
This is the trickiest!
Each \(W_i = (i, \infty)\), so the lower bound increases as \(i\to\infty\). For any real number \(x\), you can find an \(i\) such that \(x \le i\), which means \(x \notin W_i\). So:
- No real number is in *every* \(W_i\).
- The intersection is empty.
Hence:
\[
\boxed{\bigcap_{i=0}^\infty W_i = \varnothing}
\]
