## Wednesday, January 2, 2019

### Set (CIE)

$\def\D{\displaystyle}$
1 (CIE 2012, s, paper 21, question 6)
By shading the Venn diagrams below, investigate whether each of the following statements is true or false. State your conclusions clearly.
(i) $\D A\cap B = (A'\cap B)'$  

(ii) $\D X\cap Y = X'\cup Y'$ 

(iii) $\D (P\cap Q)\cup (Q\cap R) = Q\cap (P\cup R)$ 

3 (CIE 2012, w, paper 13, question 1)
(a) On the Venn diagrams below, shade the region corresponding to the set given below each Venn diagram.


(b) It is given that sets $\D E, B, S$ and $\D F$ are such that
$\D E$ = {students in a school},
$\D B =$ {students who are boys},
$\D S =$ {students in the swimming team},
$\D F =$ {students in the football team}.
Express each of the following statements in set notation.
(i) All students in the football team are boys. 
(ii) There are no students who are in both the swimming team and the football team. 

4 (CIE 2012, w, paper 21, question 2)
(a) It is given that $\D E$ is the set of integers, $\D P$ is the set of prime numbers between 10 and 50, $\D F$ is the set of multiples of 5, and $\D T$ is the set of multiples of 10. Write the following statements using set notation.
(i) There are 11 prime numbers between 10 and 50. 
(ii) 18 is not a multiple of 5. 
(iii) All multiples of 10 are multiples of 5. 
(b) (i) In the Venn diagram below shade the region that represents $\D (A'\cap B) \cup (A\cap B').$ 

(ii) In the Venn diagram below shade the region that represents $\D Q\cap (R\cup S' ).$ 

5 (CIE 2013, s, paper 12, question 1)

The Venn diagram shows the universal set $\D E$, the set $\D A$ and the set $\D B.$ Given that $\D n(B ) = 5, n(A') = 10$
and $\D n(E) = 26,$ find
(i) $\D n(A \cap B),$ 
(ii) $\D n(A),$ 
(iii) $\D n(B' \cap A).$ 

6 (CIE 2013, s, paper 21, question 9)
It is given that $\D x \in R$ and that 
$\D E = {x : − 5 < x < 12},$
$\D S = {x : 5x + 24 > x^2},$
$\D T = {x : 2x + 7 > 15}.$
Find the values of $\D x$ such that
(i) $\D x \in S ,$ 
(ii) $\D x \in S\cup T ,$ 
(iii) $\D x \in (S\cap T )'.$ 

7 (CIE 2013, w, paper 11, question 4)
The sets $\D A$ and $\D B$ are such that
$\D A=\{ x:\cos x=\frac{1}{2} , 0^{\circ}\le x\le 620^{\circ},\}$
$\D B=\{ x:\tan x=\sqrt{3} , 0^{\circ}\le x\le 620^{\circ},\}$
(i) Find $\D n(A).$ 
(ii) Find $\D n(B).$ 
(iii) Find the elements of $\D A \cup B.$ 
(iv) Find the elements of $\D A \cap B.$ 

8 (CIE 2013, w, paper 23, question 5)
(a) (i) In the Venn diagram below shade the region that represents $\D (A\cup B)'$ 

(ii) In the Venn diagram below shade the region that represents $\D P\cap Q\cap R'.$ 

(b) Express, in set notation, the set represented by the shaded region. 

(c) The universal set $\D E$ and the sets $\D V$ and $\D W$ are such that $\D n(E) = 40, n(V ) = 18$ and $\D n(W) = 14.$ Given that $\D n(V\cap W) = x$ and $\D n((V\cup W)') = 3x$ find the value of $\D x.$

9 (CIE 2014, s, paper 11, question 3)
(a) On the Venn diagrams below, shade the regions indicated.


(b) Sets $\D P$ and $\D Q$ are such that $\D P=\{x:x^2+2x=0\},$ and $\D Q=\{x:x^2+2x+7=0\},$ where $\D x\in R.$
(i) Find $\D n(P).$ 
(ii) Find $\D n(Q).$ 

10 (CIE 2014, s, paper 12, question 2)
(a) On the Venn diagrams below, draw sets $\D A$ and $\D B$ as indicated.
(i)
(ii)

(b) The universal set $\D E$ and sets $\D P$ and $\D Q$ are such that $\D n(E) = 20, n(P \cup Q) = 15, n(P) = 13$  and $\D n(P \cap Q) = 4.$ Find
(i) $\D n(Q),$ 
(ii) $\D n(P \cup Q)',$ 
(iii) $\D n(P \cap Q').$ 

11 (CIE 2014, s, paper 23, question 4)
(a) Illustrate the following statements using the Venn diagrams below.
(i) $\D A \cup B = A$ (ii) $\D A \cap B \cap C = \emptyset.$ 

(b) It is given that $\D E$ is the set of integers between 1 and 100 inclusive. $\D S$ and $\D C$ are subsets of $\D E$, where $\D S$ is the set of square numbers and $\D C$ is the set of cube numbers. Write the following statements using set notation.
(i) 50 is not a cube number. 
(ii) 64 is both a square number and a cube number. 
(iii) There are 90 integers between 1 and 100 inclusive which are not square numbers. 

12 (CIE 2014, w, paper 13, question 3)
The universal set $\D E$ is the set of real numbers. Sets $\D A, B$ and $\D C$ are such that
$\D A = \{x:x^2+5x+6=0\},$
$\D B = \{x:(x-3)(x+2)(x+1)=0\},$
$\D C = \{x:x^2+x+3=0\}.$
(i) State the value of each of $\D n(A), n(B)$ and $\D n(C).$ 
(ii) List the elements in the set $\D A \cup B.$ 
(iii) List the elements in the set $\D A \cap B.$ 
(iv) Describe the set $\D C'.$ 

13 (CIE 2014, w, paper 21, question 1)
(a) On each of the Venn diagrams below shade the region which represents the given set.

(b) In a year group of 98 pupils, $\D F$ is the set of pupils who play football and $\D H$ is the set of pupils who play hockey. There are 60 pupils who play football and 50 pupils who play hockey. The number that play both sports is $\D x$ and the number that play neither is $\D 30 - 2x.$ Find the value of $\D x.$