Worksheet 2: Set Operations - Find unions and intersections of sets A, B, C, D, E, F within universal set U = {1,2,...,9}.

We are given the universal set U = {1, 2, 3, 4, 5, 6, 7, 8, 9} and several subsets:

- 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} *(not needed for this problem)*
- F = {1, 5, 9}

We are to find:

> (a) A ∪ B and A ∩ B
> (b) A ∪ C and A ∩ C
> (c) D ∪ F and D ∩ F

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🔹 Definitions Recap:

- Union (X ∪ Y): All elements that are in X or Y or both.
- Intersection (X ∩ Y): Only elements that are in both X and Y.

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## ✔ (a) A ∪ B and A ∩ B

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

→ A ∪ B: Combine all unique elements from both sets.
= {1, 2, 3, 4, 5, 6, 7}

→ A ∩ B: Elements common to both A and B.
Common elements: 4, 5
= {4, 5}

✔ Answer (a):
A ∪ B = {1, 2, 3, 4, 5, 6, 7}
A ∩ B = {4, 5}

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## ✔ (b) A ∪ C and A ∩ C

A = {1, 2, 3, 4, 5}
C = {5, 6, 7, 8, 9}

→ A ∪ C: All elements in either A or C.
= {1, 2, 3, 4, 5, 6, 7, 8, 9} → which is the universal set U.

→ A ∩ C: Elements common to both A and C.
Only common element: 5
= {5}

✔ Answer (b):
A ∪ C = {1, 2, 3, 4, 5, 6, 7, 8, 9}
A ∩ C = {5}

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## ✔ (c) D ∪ F and D ∩ F

D = {1, 3, 5, 7, 9}
F = {1, 5, 9}

→ D ∪ F: All elements in D or F. Since F is a subset of D, union is just D.
= {1, 3, 5, 7, 9}

→ D ∩ F: Elements common to both D and F.
All elements of F are in D → so intersection is F.
= {1, 5, 9}

✔ Answer (c):
D ∪ F = {1, 3, 5, 7, 9}
D ∩ F = {1, 5, 9}

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## 📝 Final Answers:

(a)
A ∪ B = {1, 2, 3, 4, 5, 6, 7}
A ∩ B = {4, 5}

(b)
A ∪ C = {1, 2, 3, 4, 5, 6, 7, 8, 9}
A ∩ C = {5}

(c)
D ∪ F = {1, 3, 5, 7, 9}
D ∩ F = {1, 5, 9}

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