INTRODUCTION TO MATHEMATICAL METHODS

QUIZ ONE JULY INTAKE 2023

DATE : 11th August, 2023

INSTRUCTIONS

1. Duration: 30 min.
2. Write your Computer Number on each answer paper used. Answer all questions and show all the necessary work to earn full marks.

QUESTION ONE

(a) Write set $B=\{a_n:a_{n+1}=3a_n,n\in N,a_1=2\}$ in Roster form.
Solution:

• $a_1=2$
• $a_2=3\times 2=6$
• $a_3=3\times 6=18$
• $a_4=3\times 18=54$
• ...
  So $B=\{2,6,18,54,\dots \}$.

(b) Write set $D=\{5,25,125,625,\dots \}$ in set-builder form.
Solution:
$D=\{5^n\mid n\in N\}$.

(c) If $A$ is empty, show $P\left(P\right(P\left(A\right)\left)\right)$ has 4 elements.
Solution:

• $A=\varnothing$
• $P\left(A\right)=\left\{\varnothing \right\}$ (1 element)
• $P\left(P\right(A\left)\right)=P\left(\right\{\varnothing \left\}\right)=\{\varnothing ,\{\varnothing \left\}\right\}$ (2 elements)
• $P\left(P\right(P\left(A\right)\left)\right)=P\left(\right\{\varnothing ,\left\{\varnothing \right\}\left\}\right)=\{\varnothing ,\{\varnothing \},\{\left\{\varnothing \right\}\},\{\varnothing ,\left\{\varnothing \right\}\left\}\right\}$ (4 elements).

QUESTION TWO

(a) Express $12.3231$ as $\frac{a}{b}$ (coprime).
Solution:
$12.3231=\frac{123231}{10000}$, GCD(123231,10000)=1.
Final Answer: $\frac{123231}{10000}$

(b) Survey: 200 students, chatrooms: skateboarding (S), bicycling (B), college (C).

• $n\left(S\right)=90$, $n\left(B\right)=50$, $n\left(C\right)=70$
• $n(S\cap C)=15$, $n(B\cap C)=12$, $n(S\cap B)=25$, $n(S\cap B\cap C)=10$
• None: ?

(i) $n(S\cup B)$

Solution:

$n(S\cup B)=n\left(S\right)+n\left(B\right)-n(S\cap B)$

$=90+50-25$

$=115$

$\mathrm{<strong>(ii)\; None</strong>}$

$\mathrm{SOLUTION}$

$n(S\cup B\cup C)=n\left(S\right)+n\left(B\right)+n\left(C\right)-n(S\cap B)-n(S\cap C)-n(B\cap C)+n(S\cap B\cap C)$ $=90+50+70-25-15-12+10=168$

None: $200-168=32$.

@Dr. Microbiota

END OF SOLUTIONS