Worksheet on set operations including union, intersection, and complement with a problem involving the letters of words.

Let's solve the problem step by step.

We are given three sets:

- $ A = \{e, u, w, x, y\} $
- $ B $: the letters of the word 'weather'
- $ C = \{w, i, n\} $

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Step 1: Determine Set $ B $

The word "weather" consists of the letters:
w, e, a, t, h, e, r

But in set notation, we only include unique elements (no duplicates), so:

$$
B = \{w, e, a, t, h, r\}
$$

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Now let’s solve each part.

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a) $ A \cup B $ — Union of A and B

The union of two sets includes all elements that are in either set (without repetition).

- $ A = \{e, u, w, x, y\} $
- $ B = \{w, e, a, t, h, r\} $

Combine all unique elements from both sets:

- From A: $ e, u, w, x, y $
- From B: $ a, t, h, r $ (note: $ e $ and $ w $ already in A)

So,

$$
A \cup B = \{e, u, w, x, y, a, t, h, r\}
$$

✔ Final answer for (a):
$$
\boxed{\{a, e, h, r, t, u, w, x, y\}}
$$

*(Note: Sets are unordered, so order doesn't matter. We can list them alphabetically for clarity.)*

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b) $ A \cap B $ — Intersection of A and B

The intersection includes only elements that are in both sets.

- $ A = \{e, u, w, x, y\} $
- $ B = \{w, e, a, t, h, r\} $

Common elements:
- $ e $ → in both
- $ w $ → in both

So,

$$
A \cap B = \{e, w\}
$$

✔ Final answer for (b):
$$
\boxed{\{e, w\}}
$$

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c) $ A \cup C $ — Union of A and C

- $ A = \{e, u, w, x, y\} $
- $ C = \{w, i, n\} $

Combine all unique elements:

- From A: $ e, u, w, x, y $
- From C: $ i, n $ (note: $ w $ is already in A)

So,

$$
A \cup C = \{e, u, w, x, y, i, n\}
$$

✔ Final answer for (c):
$$
\boxed{\{e, i, n, u, w, x, y\}}
$$

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✔ Final Answers:

a) $ A \cup B = \{a, e, h, r, t, u, w, x, y\} $
b) $ A \cap B = \{e, w\} $
c) $ A \cup C = \{e, i, n, u, w, x, y\} $

Let me know if you'd like a Venn diagram explanation or further clarification!