Solve set theory problems using the provided Venn diagram.

We are given a Venn diagram with three sets: A, B, and C. The numbers in the regions represent elements (or counts) in those specific regions.

Let’s first identify what each region represents by listing the elements in each set and their intersections.

From the diagram:

- Only A (not B or C): 14, 8
- Only B (not A or C): 5
- Only C (not A or B): 10
- A ∩ B only (not C): 3, 16
- A ∩ C only (not B): 17, 15
- B ∩ C only (not A): 12, 13
- A ∩ B ∩ C: 9
- Outside all sets: 7, 18

So we can define the sets as:

- A = {3, 8, 9, 14, 15, 16, 17}
- B = {3, 5, 9, 12, 13, 16}
- C = {9, 10, 12, 13, 15, 17}

Also, universal set U includes all numbers shown:
U = {3, 5, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18}

Now let’s solve each problem one by one.

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1) A' ∪ (B ∩ C)'

First, find A’ = U - A = {5, 7, 10, 12, 13, 18}

Then, B ∩ C = {9, 12, 13} → so (B ∩ C)’ = U - {9,12,13} = {3,5,7,8,10,14,15,16,17,18}

Now, A’ ∪ (B ∩ C)’ = {5,7,10,12,13,18} ∪ {3,5,7,8,10,14,15,16,17,18} = {3,5,7,8,10,12,13,14,15,16,17,18}

✔ Answer: {3,5,7,8,10,12,13,14,15,16,17,18}

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2) (B ∪ C)' ∪ A

B ∪ C = {3,5,9,10,12,13,15,16,17} → (B ∪ C)' = U - that = {7,8,14,18}

A = {3,8,9,14,15,16,17}

So union: {7,8,14,18} ∪ {3,8,9,14,15,16,17} = {3,7,8,9,14,15,16,17,18}

✔ Answer: {3,7,8,9,14,15,16,17,18}

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3) C ∪ (A - B)

A - B = elements in A but not in B = {8,14,15,17} (since 3,9,16 are in B)

C = {9,10,12,13,15,17}

Union: {8,14,15,17} ∪ {9,10,12,13,15,17} = {8,9,10,12,13,14,15,17}

✔ Answer: {8,9,10,12,13,14,15,17}

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4) A - (B ∩ C)

B ∩ C = {9,12,13}

A = {3,8,9,14,15,16,17}

So remove 9 from A → {3,8,14,15,16,17}

✔ Answer: {3,8,14,15,16,17}

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5) C - (A ∪ B)

A ∪ B = {3,5,8,9,12,13,14,15,16,17}

C = {9,10,12,13,15,17}

Remove elements of A ∪ B from C → only 10 remains (since 9,12,13,15,17 are in A ∪ B)

✔ Answer: {10}

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6) C ∩ (A ∩ B)'

First, A ∩ B = {3,9,16}

So (A ∩ B)' = U - {3,9,16} = {5,7,8,10,12,13,14,15,17,18}

C = {9,10,12,13,15,17}

Intersection: common elements → {10,12,13,15,17} (since 9 is excluded)

✔ Answer: {10,12,13,15,17}

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7) A' ∩ (B - C)'

First, A’ = {5,7,10,12,13,18}

B - C = elements in B but not in C = {3,5,16} (since 9,12,13 are in C)

So (B - C)' = U - {3,5,16} = {7,8,9,10,12,13,14,15,17,18}

Now intersect with A’: {5,7,10,12,13,18} ∩ {7,8,9,10,12,13,14,15,17,18} = {7,10,12,13,18}

✔ Answer: {7,10,12,13,18}

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8) (A ∪ B)' ∪ C

From earlier: (A ∪ B)' = {7,8,14,18}

C = {9,10,12,13,15,17}

Union: {7,8,14,18} ∪ {9,10,12,13,15,17} = {7,8,9,10,12,13,14,15,17,18}

✔ Answer: {7,8,9,10,12,13,14,15,17,18}

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9) (A ∪ C) ∩ B

A ∪ C = {3,8,9,10,12,13,14,15,16,17}

B = {3,5,9,12,13,16}

Intersection: common elements → {3,9,12,13,16}

✔ Answer: {3,9,12,13,16}

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10) (A ∪ B)' ∪ C'

(A ∪ B)' = {7,8,14,18} (from before)

C' = U - C = {3,5,7,8,14,16,18}

Union: {7,8,14,18} ∪ {3,5,7,8,14,16,18} = {3,5,7,8,14,16,18}

✔ Answer: {3,5,7,8,14,16,18}

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## ✔ Final Answers:

1) {3,5,7,8,10,12,13,14,15,16,17,18}
2) {3,7,8,9,14,15,16,17,18}
3) {8,9,10,12,13,14,15,17}
4) {3,8,14,15,16,17}
5) {10}
6) {10,12,13,15,17}
7) {7,10,12,13,18}
8) {7,8,9,10,12,13,14,15,17,18}
9) {3,9,12,13,16}
10) {3,5,7,8,14,16,18}

You can write these in any order — since sets are unordered — and you may omit braces if the worksheet expects just listing numbers.

Let me know if you want them boxed or formatted for submission!