Sets and Venn Diagram topical question[Pearson Edexcel Mathematics B (4MB1) 2014-2016]

Have you already known...... 

 1. Set Notation A set is a collection of distinct elements: 

\(A = \{2,4,6\}\). 

 Universal set \(\mathcal{E}\) (or \(U\)): all elements under consideration. 

 Empty set \(\varnothing\): no elements.

 Cardinality \(n(A)\): number of elements in \(A\). 

 2. Subsets and Proper Subsets 

{Subset} \(A \subseteq B\): every element of \(A\) is in \(B\). 

{Proper subset} \(A \subset B\): \(A \subseteq B\) and \(A \neq B\). \(\varnothing\) is a subset of every set. 

 3. Set Operations \[ \begin{array}{lll} \text{Union} & A \cup B & \text{elements in } A \text{ or } B \text{ (or both)} \\ \text{Intersection} & A \cap B & \text{elements in both } A \text{ and } B \\ \text{Complement} & A' \text{ or } A^c & \text{elements not in } A \text{ (within } \mathcal{E}) \\ \text{Difference} & A \setminus B & \text{elements in } A \text{ but not in } B \end{array} \] 

 4. Venn Diagrams 

 Rectangle represents universal set

 \(\mathcal{E}\); circles represent sets. 

 Overlapping region = intersection. 

 Outside all circles but inside rectangle = complement of union. 

** Start filling from the innermost intersection outward. **

5. Solving Problems with Venn Diagrams Identify \(\mathcal{E}\) and given sets. 

 Draw diagram, label regions. Use total counts to form equations. Solve for unknowns. 

 {Example:} 38 bus, 27 train, 5 neither, total 50, both \(=x\). \[ (38-x)+(27-x)+x+5 = 50 \;\Longrightarrow\; x = 20. \] 

 6. Probability in Sets If an element is chosen at random from \(\mathcal{E}\) (equally likely outcomes): \[ P(A) = \frac{n(A)}{n(\mathcal{E})}. \] 

{Key formulas:} \[ \begin{aligned} P(A \cup B) &= P(A) + P(B) - P(A \cap B) \\ P(A') &= 1 - P(A) \\ P(A \cap B) &= 0 \quad \text{if } A \text{ and } B \text{ are disjoint (mutually exclusive)}. \end{aligned} \] {Example:} Only bus = \(18\), total = \(50\) \(\Rightarrow\) \(P(\text{only bus}) = \frac{18}{50} = \frac{9}{25}\). 

Are you ready to practise topical questions? Let's do it !

1	(Edexcel/Math B/2014/jan/paper01/q6) \[ \mathcal{E} = \{\text{positive integers } < 15\} \] \[ A = \{\text{prime numbers}\} \] \[ B = \{\text{even numbers}\} \]  Find: (a)\( B' \) -----[1] (b)\( A \cap B \) -----[1]
2	(Edexcel/Math B/2014/jan/paper01/q17) The Venn diagram below shows information about the elements in the three sets \(A\), \(B\) and \(C\). Write down the elements in (a) \( A \cap B \cap C \)-----[1] (b) \( (A \cup B) \cap C \)-----[1] Find (c) \( n[A \cup (B \cap C)] \)-----[1]
3	(Edexcel/Math B/2014/jan/paper01R/q7) \(A\) and \(B\) are two sets. Given that \(n(A) = 20\), \(n(B) = 15\) and \(n(A \cap B) = 6\), find \(n(A \cup B)\).-----[2]
4	(Edexcel/Math B/2014/jan/paper02R/q7) A sports club has 80 members.  For the three activities Swimming (\(S\)), Cycling (\(C\)) and Running (\(R\)) 8 members take part in all three activities. 3 members do not take part in any of the three activities. 22 members take part in only Swimming. 23 members take part in Swimming and Cycling. 19 members take part in Swimming and Running. 14 members take part in Cycling and Running.  (a) Using this information place the number of members in the appropriate subsets of the Venn diagram.-----[3] The number of members who take part in only Cycling is twice the number of members who take part in only Running. Let the number of members who take part in only Running be \(x\) and using all the given information, (b) form an equation in \(x\).-----[1] (c) Solve your equation to find the value of \(x\).-----[2] Manuel is in the set \((R \cup C)' \cap S\). (d) Write down which of the three activities Manuel takes part in.-----[1] (e) Write down  (i) \(n(C)\),  (ii) \(n(S \cap (R \cup C))\).-----[2]  A member of the sports club is to be chosen at random. Given that this member takes part in Cycling, (f) find the probability that this member also takes part in both Swimming and Running.-----[2]
5	(Edexcel/Math B/2014/june/paper01/q21) Some workers were asked how they travel to work.  Of these workers  38 travel to work by bus (\(B\)),  27 travel to work by train (\(T\)),  5 do not travel to work by bus or by train,  \(x\) travel to work by both bus and train.  (a) Using this information, complete the Venn diagram, giving your answers in terms of \(x\) where appropriate.-----[1] Given that the number of workers asked is 50,  (b) calculate the value of \(x\).-----[2]  One of the workers is chosen at random.  (c) Find the probability that this worker travels to work only by bus.-----[2]
6	(Edexcel/Math B/2014/june/paper02/q1) \[ \mathcal{E} = \{x : 2 \le x \le 10 \text{ and } x \text{ is an integer}\} \] \[ A = \{x : 3 \le x \le 8\} \] \[ B' = \{x : x \text{ is prime}\} \] \[ C = \{x : x \text{ is an even integer}\} \] List the elements of  (a) \(\mathcal{B}'\)-----[1]  (b) \(\mathcal{A} \cap C\)-----[1]  Find:  (c)  \(n([A \cap C]' \cap B')\) -----[2]
7	(Edexcel/Math B/2014/june/paper01R/q24) \[ \mathcal{E} = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \] \[ A = \{ \text{multiples of } 3 \} \] \[ B = \{ \text{odd numbers} \} \] \[ C = \{ \text{factors of } 24 \} \] Complete the Venn diagram by putting each element of \(\mathcal{E}\) in the correct subset of \(\mathcal{E}\).-----[5]
8	(Edexcel/Math B/2015/jan/paper01/q13) $A=\{w,x,y,z\}$   Write down all of the subsets of \(A\) that have exactly 2 elements.-----[3]
9	(Edexcel/Math B/2015/june/paper01/q16) \[ \mathcal{E} = \{a, b, c, d, e, f, g, h, i, j\} \] \[ A = \{a, b, e, f\} \] \[ B = \{b, c, d, e, g, h\} \] \[ C = \{e, f, g, h, i, j\} \] List the elements of the sets  (a) \(A \cap B \cap C\)-----[1]  (b) \((A \cup B)'\)-----[1]  (c) \(B \cap C'\)-----[1]
10	(Edexcel/Math B/2016/jan/paper01/q14) \[ \mathcal{E} = \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\} \] \[ A = \{x : 0 lt; x < 11\} \] \[ B = \{x : 3 < x < 10\} \] \[ C = \{x : 0 \le x \le 5\} \] List the elements of  (a) \(A'\)-----[1]  (b) \(B \cup C\) -----[1]  (c) \(A \cap B\) -----[1]
	to be continued...

Answer Key for Sets and Venn Diagram (2014-2016)

1.   (a)  $1,3,5,7,9,11,13 $ (b) $2$  
2.   (a) $g, y$ (b) $g, w, y, z$ (c) 7
3.   $29 \quad$
4.   (a) $15,11$ and $6$ (b) 80 (c) $x=5 \quad$ (d) swimming (e)(i) 39 (ii) 34 (f) numerator 8 , denominator 39
5.   (a) $ 38-x, 27-x, 5$ (b) $ x=20$ (c) $ \frac{18}{50} $
6.   (a) $\{4,6 , 8,9,10\}$ (b) $\{4,6,8\}$ (c) $(A \cap C)^{\prime} \cap B^{\prime}=\{9,10\} ; 2$
7.   diagram
8.   $\{w, x\},\{w, y\},\{w, z\},\{x, y\},\{x, z\}$, $\{y, z\} \quad$ 
9.   (a) e (b) i,j $\quad$ (c) $b, c, d$ 
10.   (a) $0,11,12$ (b) $0,1,2,3,4,5,6,7,8,9$ (c) $4,5,6,7,8,9$
11.   (a) $ 8$ (b) $ 1,2,3,4,6,8,12,24 $ (c) $\{3,6,12,14\}$
12.   $(A \cup B)^{\prime}=\left\{f, h, j \right\}$
13.   (a) $B={p, r, q, s}$ (b) $C={p,t,q,s}$ (c) ${r,t}$ 
14.   (a) $19-x, 35-x$ (b)(i) $(19-x)+(35-x)+x+16=62 \quad$ (ii) $x=8 \quad$ (c) $\frac{8}{19}$
15.   (a) Who do not enjoy any of three activities. (b) $9,4,20,7,25$ (c) $x=10 \quad$ (d) (i) $46 \quad$ (ii) $23 \quad$ (iii) 25
16.   show 
17.   (a) $10,45$ and $8,25-x, 13-x$ (b) c's six terms $=90 \quad$ (c) 11 (d)(i) 35 (ii)55
18.   (a) 2,10,9  (b) 20
19.   Shade 
20.   (a) $11,16,26,7$ (b) $18$
21.   (a) 18 (b) 10 (c) 16 
22.   (a) 38 (b) 19 (c) 34
23.   (a) $8-x, 13-x, 7-x, 16,13,9$ (b)(ii) $x=5$
24.   (a) $A \cap B \cap C$ (b) $\operatorname{C} \cap(A \cup B)^{\prime}$ (c) $ A \cap B \cap C^{\prime}$