Venn Diagram Set Notation Problems Worksheet with Answer Key

Let's solve the Venn diagram problems step by step based on the provided image and answer key. The goal is to understand how set notation operations (union, intersection, complement, difference) are applied using the given elements in each diagram.

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Problem 1: Sets A and B

Venn Diagram:
- Universal set $ U $ contains all elements.
- Set $ A $: {5, 10, 35, 20}
- Set $ B $: {10, 15, 35, 25}
- Intersection $ A \cap B $: {10, 35}
- Outside both: {40, 25}

We are given:
- $ A \cup B' = \{5, 20, 10, 35, 40, 25\} $ ✔ (already solved)

Now solve:

#### 1. $ A' - B $

This means: Elements not in A, but not in B either? Wait — let’s clarify:

- $ A' $: Complement of A → All elements not in A
- Then $ A' - B $: From $ A' $, remove elements that are in $ B $

So:
- $ A' = U - A $
- $ U = \{5, 10, 15, 20, 25, 35, 40\} $
- $ A = \{5, 10, 20, 35\} $
- So $ A' = \{15, 25, 40\} $
- Now subtract $ B $ from $ A' $: $ A' - B = \{15, 25, 40\} - \{10, 15, 25, 35\} = \{40\} $

Wait! But the answer key says: $ A' - B = \{40, 25\} $

That can't be right unless there's a mistake.

Wait — let's recheck.

But wait — look at the diagram:

- $ A = \{5, 10, 20, 35\} $
- $ B = \{10, 15, 25, 35\} $
- $ U = \{5, 10, 15, 20, 25, 35, 40\} $

So $ A' = \{15, 25, 40\} $

Now $ A' - B = \{15, 25, 40\} - \{10, 15, 25, 35\} = \{40\} $

But the key says $ \{40, 25\} $. That suggests $ 25 $ is not in $ B $? But it is!

Wait — maybe I misread the diagram.

Look again:

In the top-left diagram:

- Only A: 5, 20
- Only B: 15, 25
- Both: 10, 35
- Outside: 40

So:
- $ A = \{5, 10, 20, 35\} $
- $ B = \{10, 15, 25, 35\} $
- $ U = \{5, 10, 15, 20, 25, 35, 40\} $

So $ A' = \{15, 25, 40\} $

$ B = \{10, 15, 25, 35\} $

So $ A' - B = \{15, 25, 40\} - \{10, 15, 25, 35\} = \{40\} $

But the answer key says $ \{40, 25\} $ — this is incorrect?

Wait — perhaps the notation is different.

Wait — look at the first one:

They say: $ A \cup B' = \{5, 20, 10, 35, 40, 25\} $

Let’s verify that:

- $ B' = U - B = \{5, 20, 40\} $
- $ A \cup B' = \{5, 10, 20, 35\} \cup \{5, 20, 40\} = \{5, 10, 20, 35, 40\} $

But they list $ \{5, 20, 10, 35, 40, 25\} $ — includes 25?

But 25 is in $ B $, so not in $ B' $, so shouldn't be in $ A \cup B' $

So their first answer is wrong?

Wait — no! Let's double-check.

$ B = \{10, 15, 25, 35\} $

So $ B' = U - B = \{5, 20, 40\} $

$ A = \{5, 10, 20, 35\} $

So $ A \cup B' = \{5, 10, 20, 35\} \cup \{5, 20, 40\} = \{5, 10, 20, 35, 40\} $

But the answer key says: $ \{5, 20, 10, 35, 40, 25\} $ — includes 25.

But 25 is in $ B $, so not in $ B' $, and not in $ A $, so not in union.

So this is an error.

Wait — unless the universal set includes more?

Wait — the numbers outside the circles are 40 and 25? But 25 is in B.

No — look at the diagram:

- Left side: only A: 5, 20
- Right side: only B: 15, 25
- Intersection: 10, 35
- Outside both: 40

So $ U = \{5, 10, 15, 20, 25, 35, 40\} $

Yes.

So $ B' = U - B = \{5, 20, 40\} $

$ A \cup B' = \{5, 10, 20, 35\} \cup \{5, 20, 40\} = \{5, 10, 20, 35, 40\} $

But the key says $ \{5, 20, 10, 35, 40, 25\} $ — includes 25, which is not correct.

So the first answer is wrong.

But then why does it include 25?

Unless... is 25 outside? No — it's in B.

Wait — maybe the diagram has a typo?

Wait — no. The number 25 is inside B, not outside.

So $ B' $ should not include 25.

So the first answer is incorrect.

But let's continue with what’s written.

Maybe we’re supposed to trust the answer key?

Wait — look at the next one: $ A' \cap B' $

Answer key says: $ \{40, 25\} $

Let’s compute:

- $ A' = U - A = \{15, 25, 40\} $
- $ B' = U - B = \{5, 20, 40\} $
- $ A' \cap B' = \{15, 25, 40\} \cap \{5, 20, 40\} = \{40\} $

But the key says $ \{40, 25\} $ — again, 25 is not in $ B' $

So clearly the answer key has errors.

Wait — unless the universal set includes 25 in both A and B? But it’s only in B.

Wait — maybe the number 25 is outside both sets, but placed in B?

No — the diagram shows:

- B circle has: 15, 25, and 10, 35 (shared)
- So 25 is in B only.

So $ B' $ excludes 25.

So the answer key is wrong for the first problem.

But let’s check the other diagrams.

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Problem 2: Sets C and D

Diagram:

- C only: 8, 21
- D only: 14, 26
- Both: 12, 7, 15
- Outside: 27, 4, 32

So:
- $ C = \{8, 12, 21, 7, 15\} $
- $ D = \{12, 14, 7, 15, 26\} $
- $ U = \{8, 12, 14, 15, 18, 21, 26, 27, 32, 4\} $ — wait, numbers: 8, 12, 21, 14, 26, 27, 4, 32, 7, 15

So $ U = \{4, 7, 8, 12, 14, 15, 21, 26, 27, 32\} $

Now:

#### $ C \cap D' $

- $ D' = U - D $
- $ D = \{12, 14, 7, 15, 26\} $
- $ D' = \{4, 8, 21, 27, 32\} $
- $ C = \{8, 12, 21, 7, 15\} $
- $ C \cap D' = \{8, 21\} $ — because 8 and 21 are in C and in D'

Answer key says: $ \{8, 12, 21\} $ — includes 12

But 12 is in D, so not in D', so not in $ C \cap D' $

So error — should be $ \{8, 21\} $

But key says $ \{8, 12, 21\} $

Again, error.

Next: $ D' - C' $

Answer key: $ \{8, 12, 21\} $

Let’s compute:

- $ D' = \{4, 8, 21, 27, 32\} $
- $ C' = U - C = \{4, 14, 26, 27, 32\} $
- $ D' - C' = \{4, 8, 21, 27, 32\} - \{4, 14, 26, 27, 32\} = \{8, 21\} $

Key says $ \{8, 12, 21\} $ — again, 12 not in D', so not possible.

So multiple errors in the key.

Wait — perhaps the diagram is interpreted differently.

Wait — look at the numbers:

- In C: 8, 21, and shared: 12, 7, 15 → so C = {8, 12, 21, 7, 15}
- In D: 14, 26, and shared: 12, 7, 15 → D = {12, 14, 7, 15, 26}
- Outside: 27, 4, 32

So U = {4, 7, 8, 12, 14, 15, 21, 26, 27, 32}

Now $ C \cap D' $:

- D' = U - D = {4, 8, 21, 27, 32}
- C ∩ D' = {8, 21} — since 8 and 21 are in C and in D'

But key says {8, 12, 21} — 12 is in D, so not in D'

So incorrect

Similarly, $ D' - C' $:

- C' = U - C = {4, 14, 26, 27, 32}
- D' = {4, 8, 21, 27, 32}
- D' - C' = {8, 21} — again, key says {8, 12, 21} — impossible

So likely the answer key is wrong.

But let’s move to the third diagram.

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Problem 3: Sets E and F

Diagram:

- E only: 19, 13, 6
- F only: 22, 8, 6
- Both: 11, 9
- Outside: 85, 17

Wait — 6 is in both E and F? But it’s listed under E only and F only? No.

Actually:
- E only: 19, 13, 6
- F only: 22, 8, 6 — wait, 6 is in both?
- But 6 is listed in both "only" sections? That doesn’t make sense.

Wait — the diagram shows:

- E circle: 19, 13, 6, and intersection: 11, 9
- F circle: 22, 8, 6, and intersection: 11, 9
- Outside: 85, 17

So:
- E = {19, 13, 6, 11, 9}
- F = {22, 8, 6, 11, 9}
- Intersection: {11, 9, 6}? But 6 is in both?

But in the diagram, 6 is in both circles, but not in intersection? Or is it?

Wait — if 6 is in both E and F, it should be in the intersection.

But it’s shown separately in both “only” regions? That would be a mistake.

But looking at the layout:

- Left side (E): 19, 13, 6
- Right side (F): 22, 8, 6
- Middle: 11, 9

So 6 is in both E and F, but not in the middle — that means it’s in both but not in the intersection? That’s impossible.

Unless the diagram has a mistake.

But typically, elements in both sets go in the intersection.

So likely, 6 is in the intersection, but it’s listed twice.

But in the diagram, it’s outside the intersection — so probably it's in both, but not labeled in the overlap.

But then it should be in the overlap.

Perhaps it's a typo.

Assume that:
- E = {19, 13, 6, 11, 9}
- F = {22, 8, 6, 11, 9}
- So intersection = {6, 11, 9}
- But the diagram shows only 11 and 9 in the middle — so 6 is missing?

But then where is 6?

Wait — maybe the number 6 is in the intersection, but it’s written in both sides by mistake.

Alternatively, perhaps the diagram intends that 6 is in both, so in intersection.

But it’s not in the overlap region.

This is confusing.

But look at the answer key:

Given: $ E' \cup F' = \{19,2,13,6,22,8,85,17,6\} $ — wait, 2? Where is 2?

The numbers are: 19, 13, 6, 22, 8, 11, 9, 85, 17

No 2.

So likely typo.

But the key says $ E' \cup F' = \{19,2,13,6,22,8,85,17,6\} $ — has duplicate 6 and extra 2.

Clearly wrong.

Then $ E \cap F' = \{19,2,13,6\} $ — again, 2 not in diagram.

And $ E' \cap F' = \{85,17,6\} $

But let’s compute properly.

Assume:

- U = {19, 13, 6, 11, 9, 22, 8, 85, 17}
- E = {19, 13, 6, 11, 9}
- F = {22, 8, 6, 11, 9}
- So E ∩ F = {6, 11, 9}
- E' = U - E = {22, 8, 85, 17}
- F' = U - F = {19, 13, 85, 17}
- E' ∪ F' = {19, 13, 22, 8, 85, 17} — no 6, no 2
- But key says: {19,2,13,6,22,8,85,17,6} — includes 2, 6, duplicate 6 — wrong

So clearly, the answer key is full of errors.

Let’s try the last one.

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Problem 4: Sets B and C

Diagram:

- B only: 24, 33
- C only: 29, 7, 77
- Both: 3, 16, 1
- Outside: 12

So:
- B = {24, 33, 3, 16, 1}
- C = {29, 7, 77, 3, 16, 1}
- U = {1, 3, 7, 12, 16, 24, 29, 33, 77}

Now:

#### $ B' \cap C $

- B' = U - B = {7, 12, 29, 77}
- C = {1, 3, 7, 16, 29, 77}
- $ B' \cap C = \{7, 29, 77\} $

But key says: {29,7} — missing 77

So error

Next: $ B' \cap C' $

- C' = U - C = {12, 24, 33}
- B' = {7, 12, 29, 77}
- $ B' \cap C' = \{12\} $

But key says: {12,77} — 77 is in C, so not in C' — error

Last: $ B' - C $

- B' = {7, 12, 29, 77}
- C = {1, 3, 7, 16, 29, 77}
- B' - C = {12} — since 7, 29, 77 are in C

But key says: {12,77} — again, 77 is in C, so not in difference

So all answers in the key are wrong.

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Conclusion: The Answer Key Provided is Incorrect

There are multiple errors in the answer key. For example:

- In Problem 1: $ A \cup B' $ should be {5, 10, 20, 35, 40}, not including 25
- $ A' \cap B' $ should be {40}, not {40, 25}
- In Problem 2: $ C \cap D' $ should be {8, 21}, not {8, 12, 21}
- In Problem 4: $ B' \cap C $ should be {7, 29, 77}, not {29,7}

So the provided answer key is not reliable.

However, if you're trying to learn, here are the correct solutions:

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✔ Correct Solutions

#### Problem 1: A and B

- $ A = \{5, 10, 20, 35\} $
- $ B = \{10, 15, 25, 35\} $
- $ U = \{5, 10, 15, 20, 25, 35, 40\} $

1. $ A \cup B' = A \cup (U - B) = \{5, 10, 20, 35\} \cup \{5, 20, 40\} = \{5, 10, 20, 35, 40\} $
2. $ A' - B = (U - A) - B = \{15, 25, 40\} - \{10, 15, 25, 35\} = \{40\} $
3. $ A' \cap B' = (U - A) \cap (U - B) = \{15, 25, 40\} \cap \{5, 20, 40\} = \{40\} $

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#### Problem 2: C and D

- $ C = \{8, 12, 21, 7, 15\} $
- $ D = \{12, 14, 7, 15, 26\} $
- $ U = \{4, 7, 8, 12, 14, 15, 21, 26, 27, 32\} $

1. $ C \cap D' = C \cap (U - D) = \{8, 12, 21, 7, 15\} \cap \{4, 8, 21, 27, 32\} = \{8, 21\} $
2. $ D' - C' = (U - D) - (U - C) = \{4, 8, 21, 27, 32\} - \{4, 14, 26, 27, 32\} = \{8, 21\} $

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#### Problem 3: E and F

- $ E = \{19, 13, 6, 11, 9\} $
- $ F = \{22, 8, 6, 11, 9\} $
- $ U = \{19, 13, 6, 11, 9, 22, 8, 85, 17\} $

1. $ E' \cup F' = (U - E) \cup (U - F) = \{22, 8, 85, 17\} \cup \{19, 13, 85, 17\} = \{19, 13, 22, 8, 85, 17\} $
2. $ E \cap F' = E \cap (U - F) = \{19, 13, 6, 11, 9\} \cap \{19, 13, 85, 17\} = \{19, 13\} $
3. $ E' \cap F' = (U - E) \cap (U - F) = \{22, 8, 85, 17\} \cap \{19, 13, 85, 17\} = \{85, 17\} $

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#### Problem 4: B and C

- $ B = \{24, 33, 3, 16, 1\} $
- $ C = \{29, 7, 77, 3, 16, 1\} $
- $ U = \{1, 3, 7, 12, 16, 24, 29, 33, 77\} $

1. $ B' \cap C = (U - B) \cap C = \{7, 12, 29, 77\} \cap \{1, 3, 7, 16, 29, 77\} = \{7, 29, 77\} $
2. $ B' \cap C' = (U - B) \cap (U - C) = \{7, 12, 29, 77\} \cap \{12, 24, 33\} = \{12\} $
3. $ B' - C = \{7, 12, 29, 77\} - \{1, 3, 7, 16, 29, 77\} = \{12\} $

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🟩 Final Answer: The provided answer key contains multiple errors. Use the correct solutions above instead.