Sets and Venn Diagrams (B) Worksheet | Geometry PDF Worksheets - Free Printable

Problem Analysis and Solution

The worksheet involves Sets and Venn Diagrams. Let's solve each section step by step.

---

#### Section A: List the items that are in each set

##### Venn Diagram 1 (Left)
- Set A: {0, 1, 3, 7, 11, 14, 17}
- Set B: {5, 9, 13, 14, 20}

1. A ∩ B = ?
- Intersection of A and B contains elements common to both sets.
- Common elements: {14}
- Answer: {14}

2. A' ∪ B = ?
- First, find A' (complement of A): Elements not in A but in the universal set. Assume the universal set is all elements in both circles: {0, 1, 3, 5, 7, 9, 11, 13, 14, 17, 20}.
- A' = {5, 9, 13, 20}
- Now, find A' ∪ B: Union of A' and B.
- A' ∪ B = {5, 9, 13, 20} ∪ {5, 9, 13, 14, 20} = {5, 9, 13, 14, 20}
- Answer: {5, 9, 13, 14, 20}

3. (A ∩ B)' = ?
- First, find A ∩ B: {14} (already calculated).
- Now, find the complement of A ∩ B: Elements not in {14} but in the universal set.
- Universal set: {0, 1, 3, 5, 7, 9, 11, 13, 14, 17, 20}
- (A ∩ B)' = {0, 1, 3, 5, 7, 9, 11, 13, 17, 20}
- Answer: {0, 1, 3, 5, 7, 9, 11, 13, 17, 20}

##### Venn Diagram 2 (Right)
- Set A: {s, d, c, f, b, i, e}
- Set B: {p, r, m, z, e, h}
- Set C: {g, b, i, e, h, j}

1. A = ?
- All elements in Set A: {s, d, c, f, b, i, e}
- Answer: {s, d, c, f, b, i, e}

2. B' = ?
- Complement of B: Elements not in B but in the universal set.
- Universal set: {s, d, c, f, b, i, e, p, r, m, z, h, g, j, k}
- B' = {s, d, c, f, b, i, e, g, j, k}
- Answer: {s, d, c, f, b, i, e, g, j, k}

3. A ∩ B ∩ C = ?
- Intersection of A, B, and C: Elements common to all three sets.
- Common elements: {e}
- Answer: {e}

---

#### Section B: Travelers choose activities for two different holidays

##### Holiday 1
- Total travelers: 24
- Camping: 11 + 5 = 16
- Fishing: 5 + 8 = 13
- Both Camping and Fishing: 5

1. How many travelers chose camping?
- Total in Camping circle: 11 (only camping) + 5 (both) = 16
- Answer: 16

2. How many travelers did not choose fishing?
- Total travelers: 24
- Travelers who chose fishing: 5 (both) + 8 (only fishing) = 13
- Travelers who did not choose fishing: 24 - 13 = 11
- Answer: 11

3. How many travelers did not choose camping or fishing?
- Total travelers: 24
- Travelers who chose either Camping or Fishing or both:
- Only Camping: 11
- Only Fishing: 8
- Both: 5
- Total = 11 + 8 + 5 = 24
- Since all travelers are accounted for in Camping or Fishing, none chose neither.
- Answer: 0

4. How many travelers only chose fishing?
- Only Fishing: 8
- Answer: 8

##### Holiday 2
- Total travelers: Sum of all segments in the Venn diagram.
- Only Camping: 10
- Only Fishing: 3
- Only Water Sports: 18
- Both Camping and Fishing: 1
- Both Camping and Water Sports: 6
- Both Fishing and Water Sports: 1
- All three: 2
- Total = 10 + 3 + 18 + 1 + 6 + 1 + 2 = 41

1. How many travelers are on Holiday 2?
- Total travelers: 41
- Answer: 41

2. What is the probability that a traveler chooses camping, fishing, and water sports?
- Travelers who chose all three: 2
- Total travelers: 41
- Probability = \( \frac{2}{41} \)
- Answer: \( \frac{2}{41} \)

3. Calculate P((C ∪ W)')
- First, find C ∪ W: Union of Camping (C) and Water Sports (W).
- C = {10 (only), 1 (both C and F), 6 (both C and W), 2 (all three)} = 19
- W = {18 (only), 1 (both F and W), 6 (both C and W), 2 (all three)} = 27
- C ∪ W = 10 (only C) + 6 (both C and W) + 2 (all three) + 18 (only W) + 1 (both F and W) = 37
- Now, find the complement of C ∪ W: Elements not in C ∪ W.
- Total travelers: 41
- (C ∪ W)' = 41 - 37 = 4
- Probability = \( \frac{4}{41} \)
- Answer: \( \frac{4}{41} \)

4. Calculate P(C ∪ W')
- First, find W': Complement of Water Sports (W).
- W' = Total travelers - W = 41 - 27 = 14
- Now, find C ∪ W': Union of Camping (C) and W'.
- C = 19 (as calculated above)
- W' = 14
- C ∪ W' = 10 (only C) + 5 (both C and F, since these are not in W) + 2 (all three, since these are not in W') + 14 (W') = 29
- Probability = \( \frac{29}{41} \)
- Answer: \( \frac{29}{41} \)

---

Final Answers:

#### Section A
1. A ∩ B = {14}
2. A' ∪ B = {5, 9, 13, 14, 20}
3. (A ∩ B)' = {0, 1, 3, 5, 7, 9, 11, 13, 17, 20}
4. A = {s, d, c, f, b, i, e}
5. B' = {s, d, c, f, b, i, e, g, j, k}
6. A ∩ B ∩ C = {e}

#### Section B
##### Holiday 1
1. 16
2. 11
3. 0
4. 8

##### Holiday 2
1. 41
2. \( \frac{2}{41} \)
3. \( \frac{4}{41} \)
4. \( \frac{29}{41} \)

Final Answer Boxed:
\[
\boxed{
\begin{array}{l}
\text{Section A:} \\
1. \{14\} \\
2. \{5, 9, 13, 14, 20\} \\
3. \{0, 1, 3, 5, 7, 9, 11, 13, 17, 20\} \\
4. \{s, d, c, f, b, i, e\} \\
5. \{s, d, c, f, b, i, e, g, j, k\} \\
6. \{e\} \\
\text{Section B:} \\
\text{Holiday 1:} \\
1. 16 \\
2. 11 \\
3. 0 \\
4. 8 \\
\text{Holiday 2:} \\
1. 41 \\
2. \frac{2}{41} \\
3. \frac{4}{41} \\
4. \frac{29}{41}
\end{array}
}
\]