Venn diagram practice problems with answers for set operations.

You're correct — the image you uploaded is a completed worksheet on set operations using Venn diagrams, and all the answers are already filled in. Since you asked to “solve the problem and explain the solution,” I’ll walk through each Venn diagram and explain how the given answers were derived.

---

## 🔹 Diagram 1: Sets A and B (Numbers)

Given:

- Set A contains: {3, 7, 1, 4, 6} → elements only in A + overlap
- Set B contains: {9, 2, 8, 1, 4, 6} → elements only in B + overlap
- Overlap (A ∩ B): {1, 4, 6}

✔ A ∪ B = {1, 2, 3, 4, 5, 6, 7, 8, 9}

> Union: All elements in A or B or both.
> Combine all unique elements from both circles:
> From A: 3, 7, 1, 4, 6
> From B: 9, 2, 8, 1, 4, 6
> Unique combined: {1, 2, 3, 4, 6, 7, 8, 9} — wait, 5 is missing!

⚠️ Correction needed! The answer says `{1,2,3,4,5,6,7,8,9}` — but 5 does NOT appear anywhere in either circle. That’s likely a typo or error in the worksheet.

✔ Correct A ∪ B should be: {1,2,3,4,6,7,8,9}

---

✔ A ∩ B = {1, 4, 6}

> Intersection: Elements common to both sets → the overlapping region.
> Clearly shown in the middle: 1, 4, 6 → Correct.

---

✔ A – B = {3, 7}

> Set difference (A minus B): Elements in A but NOT in B.
> A has: 3, 7, 1, 4, 6
> Remove those also in B (1,4,6) → left with 3, 7 → Correct.

---

✔ B – A = {2, 5, 8, 9}

> Set difference (B minus A): Elements in B but NOT in A.
> B has: 9, 2, 8, 1, 4, 6
> Remove those also in A (1,4,6) → left with 9, 2, 8 — but 5 is not in B at all!

⚠️ Again, 5 is not in set B — so it shouldn’t be in B – A. This is an error.

✔ Correct B – A should be: {2, 8, 9}

---

## 🔹 Diagram 2: Sets C and D (Letters)

Given:

- Set C: {b, h, f, i, d, e} → left circle
- Set D: {a, g, c, j, k, i, d} → right circle
- Overlap (C ∩ D): {i, d}

✔ C ∪ D = {a, b, c, d, e, f, g, h, i, j, k}

> Union: All letters in either set → combine all unique letters.
> C: b, h, f, i, d, e
> D: a, g, c, j, k, i, d
> Combined unique: a, b, c, d, e, f, g, h, i, j, k → Correct.

---

✔ C ∩ D = {i, d}

> Intersection: Letters in both → overlap region → i and d → Correct.

---

✔ C – D = {b, e, f, h}

> Elements in C but not in D → remove i and d from C → left with b, e, f, h → Correct.

---

✔ D – C = {a, c, g, j, k}

> Elements in D but not in C → remove i and d from D → left with a, c, g, j, k → Correct.

---

## 🔹 Diagram 3: Sets E and F (Multiples of 3)

Given:

- Set E: {21, 9, 3, 15} → left circle
- Set F: {24, 18, 6, 12, 15} → right circle
- Overlap (E ∩ F): {15}

✔ E ∪ F = {3, 6, 9, 12, 15, 18, 21, 24}

> Union: All numbers in E or F → combine:
> E: 3, 9, 21, 15
> F: 6, 12, 18, 24, 15
> Unique: 3, 6, 9, 12, 15, 18, 21, 24 → Correct.

---

✔ E ∩ F = {15}

> Intersection: Only number in both → 15 → Correct.

---

✔ E – F = {3, 9, 21}

> Elements in E but not in F → remove 15 from E → left with 3, 9, 21 → Correct.

---

✔ F – E = {6, 12, 18, 24}

> Elements in F but not in E → remove 15 from F → left with 6, 12, 18, 24 → Correct.

---

## ✔ Final Summary:

| Operation | Answer (Worksheet) | Correction Needed? |
|------------------|----------------------------|--------------------------|
| A ∪ B | {1,2,3,4,5,6,7,8,9} | ✘ Should be {1,2,3,4,6,7,8,9} — 5 is not in any set |
| A ∩ B | {1,4,6} | ✔ Correct |
| A – B | {3,7} | ✔ Correct |
| B – A | {2,5,8,9} | ✘ Should be {2,8,9} — 5 not in B |
| C ∪ D | {a,b,c,d,e,f,g,h,i,j,k} | ✔ Correct |
| C ∩ D | {i,d} | ✔ Correct |
| C – D | {b,e,f,h} | ✔ Correct |
| D – C | {a,c,g,j,k} | ✔ Correct |
| E ∪ F | {3,6,9,12,15,18,21,24} | ✔ Correct |
| E ∩ F | {15} | ✔ Correct |
| E – F | {3,9,21} | ✔ Correct |
| F – E | {6,12,18,24} | ✔ Correct |

---

## 📝 Conclusion:

The worksheet is mostly correct, except for two errors in Diagram 1:
- 5 appears in A ∪ B and B – A, but it’s not in either set A or B.
- Likely a printing or transcription error.

✔ So if you’re submitting this as-is, note that the first diagram has incorrect answers due to the inclusion of "5".

If you’re learning from this, remember:
- Union (∪) = everything in either set.
- Intersection (∩) = only what’s in both.
- Difference (–) = what’s in the first set but NOT in the second.

Let me know if you want to practice similar problems or fix the errors!