Question 1 (5 marks) $$ A=\{2,6,7,8,9\} \quad B=\{1,2,3,5,8\} $$ List the elements of the following sets. a) $$ \mathrm{C}=\{x: x \in $$ both $$ A $$ and $$ B\} $$ b) $$ \mathrm{D}=\{x: x \in A $$ but not $$ B\} $$ c) $$ \mathrm{E}=\{x: x \in A $$ or $$ B $$ or both $$ \} $$ d) $$ \mathrm{F}=\{x: x \in B $$ but not $$ A\} $$ e) Find $$ \mathrm{n}(\mathrm{G}) $$ where $$ \mathrm{G}=\{x: x \in $$ both $$ A $$ or $$ B\} $$ Question 2 ( 5 marks) Fifty students were surveyed, and asked if they were taking a Social Science (SS), Business studies(BS) or a Physical Education (PE). 21 were taking a Social Science (SS) 26 were taking a Business Studies (BS) 28 were taking a Physical Education (PE) 9 were taking a Social Science and Business Studies 10 were taking Busiñess Studies and Physical Education 7 were taking Social Science and Physical Education 3 were taking all three 7 were taking none a) Construct a Venn diagram for the survey. b) How many students were taking SS only? c) How many students were taking PE only? d) How many students take neither Social science nor BS? e) How many students does not take SS?

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Let's solve each part of the questions step by step.

### Question 1

Given the sets:
$$
A=\{2,6,7,8,9\} \quad B=\{1,2,3,5,8\}
$$

#### a) $$ C=\{x: x \in $$ both $$ A $$ and $$ B\} $$
This set represents the intersection of sets $$ A $$ and $$ B $$.

**Elements in both $$ A $$ and $$ B $$:**
- The common elements are $$ 2 $$ and $$ 8 $$.

Thus, 
$$
C = \{2, 8\}
$$

#### b) $$ D=\{x: x \in A $$ but not $$ B\} $$
This set represents the elements in $$ A $$ that are not in $$ B $$.

**Elements in $$ A $$ but not in $$ B $$:**
- From $$ A $$: $$ 6, 7, 9 $$ are not in $$ B $$.

Thus, 
$$
D = \{6, 7, 9\}
$$

#### c) $$ E=\{x: x \in A $$ or $$ B $$ or both\} $$
This set represents the union of sets $$ A $$ and $$ B $$.

**Elements in either $$ A $$ or $$ B $$:**
- Combine all unique elements from both sets: $$ 1, 2, 3, 5, 6, 7, 8, 9 $$.

Thus, 
$$
E = \{1, 2, 3, 5, 6, 7, 8, 9\}
$$

#### d) $$ F=\{x: x \in B $$ but not $$ A\} $$
This set represents the elements in $$ B $$ that are not in $$ A $$.

**Elements in $$ B $$ but not in $$ A $$:**
- From $$ B $$: $$ 1, 3, 5 $$ are not in $$ A $$.

Thus, 
$$
F = \{1, 3, 5\}
$$

#### e) Find $$ n(G) $$ where $$ G=\{x: x \in $$ both $$ A $$ or $$ B\} $$
This set represents the union of sets $$ A $$ and $$ B $$, which we already found in part c.

The number of elements in $$ G $$ is:
$$
n(G) = |E| = 8
$$

### Question 2

Given the survey data:
- Total students = 50
- SS = 21
- BS = 26
- PE = 28
- SS and BS = 9
- BS and PE = 10
- SS and PE = 7
- All three = 3
- None = 7

#### a) Construct a Venn diagram for the survey.
To construct the Venn diagram, we need to calculate the number of students in each section of the diagram.

1. **Students taking all three subjects (SS, BS, PE)**: 3
2. **Students taking only SS**: 
   $$
   SS \text{ only} = SS - (SS \cap BS) - (SS \cap PE) + (SS \cap BS \cap PE) = 21 - 9 - 7 + 3 = 8
   $$
3. **Students taking only BS**: 
   $$
   BS \text{ only} = BS - (BS \cap SS) - (BS \cap PE) + (SS \cap BS \cap PE) = 26 - 9 - 10 + 3 = 10
   $$
4. **Students taking only PE**: 
   $$
   PE \text{ only} = PE - (PE \cap SS) - (PE \cap BS) + (SS \cap BS \cap PE) = 28 - 7 - 10 + 3 = 14
   $$
5. **Students taking SS and BS only**: 
   $$
   SS \cap BS \text{ only} = (SS \cap BS) - (SS \cap BS \cap PE) = 9 - 3 = 6
   $$
6. **Students taking BS and PE only**: 
   $$
   BS \cap PE \text{ only} = (BS \cap PE) - (SS \cap BS \cap PE) = 10 - 3 = 7
   $$
7. **Students taking SS and PE only**: 
   $$
   SS \cap PE \text{ only} = (SS \cap PE) - (SS \cap BS \cap PE) = 7 - 3 = 4
   $$

Now we can summarize:
- SS only: 8
- BS only: 10
- PE only: 14
- SS and BS only: 6
- BS and PE only: 7
- SS and PE only: 4
- All three: 3
- None: 7

#### b) How many students were taking SS only?
From our calculations:
$$
\text{SS only} = 8
$$

#### c) How many students were taking PE only?
From our calculations:
$$
\text{PE only} = 14
$$

#### d) How many students take neither Social Science nor BS?
To find this, we can calculate the total number of students taking either SS or BS:
$$
\text{Total taking SS or BS} = \text{SS only} + \text{BS only} + \text{SS and BS only} + \text{All three} = 8 + 10 + 6 + 3 = 27
$$
Thus, the number of students taking neither SS nor BS is:
$$
50 - 27 - 7 = 16
$$

#### e) How many students do not take SS?
The number of students not taking SS can be calculated as:
$$
\text{Total students} - \text{Total taking SS} = 50 - 21 = 29
$$

### Summary of Answers
1. a) $$ C = \{2, 8\} $$
   b) $$ D = \{6, 7, 9\} $$
   c) $$ E = \{1, 2, 3, 5, 6, 7, 8, 9\} $$
   d) $$ F = \{1, 3, 5\} $$
   e) $$ n(G) = 8 $$

2. a) Venn diagram constructed as described.
   b) SS only: 8
   c) PE only: 14
   d) Neither SS nor BS: 16
   e) Students not taking SS: 29
**Question 1:** - **a)** $$ C = \{2, 8\} $$ - **b)** $$ D = \{6, 7, 9\} $$ - **c)** $$ E = \{1, 2, 3, 5, 6, 7, 8, 9\} $$ - **d)** $$ F = \{1, 3, 5\} $$ - **e)** $$ n(G) = 8 $$ **Question 2:** - **a)** Venn diagram constructed with the given data. - **b)** 8 students took only Social Science. - **c)** 14 students took only Physical Education. - **d)** 16 students took neither Social Science nor Business Studies. - **e)** 29 students did not take Social Science.

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