Worksheet 2: Set Operations - Find unions and intersections of given sets.

We are given the universal set and several subsets:

- $ U = \{1, 2, 3, 4, 5, 6, 7, 8, 9\} $
- $ A = \{1, 2, 3, 4, 5\} $
- $ B = \{4, 5, 6, 7\} $
- $ C = \{5, 6, 7, 8, 9\} $
- $ D = \{1, 3, 5, 7, 9\} $
- $ E = \{2, 4, 6, 8\} $
- $ F = \{1, 5, 9\} $

We are to find:

---

(a) $ A \cup B $ and $ A \cap B $

Union $ A \cup B $:
This includes all elements that are in either $ A $ or $ B $ (or both).

- $ A = \{1, 2, 3, 4, 5\} $
- $ B = \{4, 5, 6, 7\} $

Combine all elements:
$ \{1, 2, 3, 4, 5, 6, 7\} $

So,
$$
A \cup B = \{1, 2, 3, 4, 5, 6, 7\}
$$

Intersection $ A \cap B $:
This includes elements that are in both $ A $ and $ B $.

Common elements:
- $ 4 $ is in both
- $ 5 $ is in both

So,
$$
A \cap B = \{4, 5\}
$$

---

(b) $ A \cup C $ and $ A \cap C $

Union $ A \cup C $:
- $ A = \{1, 2, 3, 4, 5\} $
- $ C = \{5, 6, 7, 8, 9\} $

Combine:
All elements from both sets:
$ \{1, 2, 3, 4, 5, 6, 7, 8, 9\} $

Note: This is actually the entire universal set $ U $.

So,
$$
A \cup C = \{1, 2, 3, 4, 5, 6, 7, 8, 9\} = U
$$

Intersection $ A \cap C $:
Elements common to both $ A $ and $ C $:
- Only $ 5 $ appears in both.

So,
$$
A \cap C = \{5\}
$$

---

(c) $ D \cup F $ and $ D \cap F $

Union $ D \cup F $:
- $ D = \{1, 3, 5, 7, 9\} $
- $ F = \{1, 5, 9\} $

Combine all elements:
$ \{1, 3, 5, 7, 9\} $ — since $ F $'s elements are already in $ D $, no new elements added.

So,
$$
D \cup F = \{1, 3, 5, 7, 9\} = D
$$

Intersection $ D \cap F $:
Elements common to both $ D $ and $ F $:
- $ 1 $, $ 5 $, $ 9 $ are in both.

So,
$$
D \cap F = \{1, 5, 9\} = F
$$

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✔ Final Answers:

(a)
- $ A \cup B = \{1, 2, 3, 4, 5, 6, 7\} $
- $ A \cap B = \{4, 5\} $

(b)
- $ A \cup C = \{1, 2, 3, 4, 5, 6, 7, 8, 9\} $
- $ A \cap C = \{5\} $

(c)
- $ D \cup F = \{1, 3, 5, 7, 9\} $
- $ D \cap F = \{1, 5, 9\} $

Let me know if you'd like a visual representation using Venn diagrams!