## 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) $\D A\cap B = (A'\cap B)'$  [2]

(ii) $\D X\cap Y = X'\cup Y'$ [2]

(iii) $\D (P\cap Q)\cup (Q\cap R) = Q\cap (P\cup R)$ [2]

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.

[2]
(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. [1]
(ii) There are no students who are in both the swimming team and the football team. [1]

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. [1]
(ii) 18 is not a multiple of 5. [1]
(iii) All multiples of 10 are multiples of 5. [1]
(b) (i) In the Venn diagram below shade the region that represents $\D (A'\cap B) \cup (A\cap B').$ [1]

(ii) In the Venn diagram below shade the region that represents $\D Q\cap (R\cup S' ).$ [1]

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) $\D n(A \cap B),$ [1]
(ii) $\D n(A),$ [1]
(iii) $\D n(B' \cap A).$ [1]

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) $\D x \in S ,$ [3]
(ii) $\D x \in S\cup T ,$ [2]
(iii) $\D x \in (S\cap T )'.$ [3]

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 $\D n(A).$ [1]
(ii) Find $\D n(B).$ [1]
(iii) Find the elements of $\D A \cup B.$ [1]
(iv) Find the elements of $\D A \cap B.$ [1]

8 (CIE 2013, w, paper 23, question 5)
(a) (i) In the Venn diagram below shade the region that represents $\D (A\cup B)'$ [1]

(ii) In the Venn diagram below shade the region that represents $\D P\cap Q\cap R'.$ [1]

(b) Express, in set notation, the set represented by the shaded region. [1]

(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.

[3]
(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 $\D n(P).$ [1]
(ii) Find $\D n(Q).$ [1]

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)
[2]
(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) $\D n(Q),$ [1]
(ii) $\D n(P \cup Q)',$ [1]
(iii) $\D n(P \cap Q').$ [1]

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) $\D A \cap B \cap C = \emptyset.$ [2]

(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. [1]
(ii) 64 is both a square number and a cube number. [1]
(iii) There are 90 integers between 1 and 100 inclusive which are not square numbers. [1]

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 $\D n(C).$ [3]
(ii) List the elements in the set $\D A \cup B.$ [1]
(iii) List the elements in the set $\D A \cap B.$ [1]
(iv) Describe the set $\D C'.$ [1]

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 $\D x.$ [3]