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Sagot :
Answer:
[tex]\noindent\rule{12cm}{0.4pt}[/tex]
a) Cardinal Notation of the Set of Students who don't like any of these subjects:
- We know that there are 10 students who don't like to study any of the subjects.
[tex]\fontsize{12}\selectfont\text{$\bullet $\ Let $U$ be the universal set representing all students in the}[/tex]
[tex]\fontsize{12}\selectfont\text{class, so $n(U)=100.$}[/tex]
[tex]\fontsize{12}\selectfont\text{$\bullet $\ Let $S$ be the set of students who like to study environmental}[/tex]
[tex]\fontsize{12}\selectfont\text{science.$}[/tex]
[tex]\fontsize{12}\selectfont\text{$\bullet$\ Let $E$ be the set of students who like to study economics. }[/tex]
[tex]\fontsize{12}\selectfont\text{$\bullet$\ So, the set of students who don't like any of these subjects is}[/tex]
[tex]\fontsize{12}\selectfont\text{denoted by $(SUE)'.$}[/tex]
Thus the cardinal notation of this set is:
[tex]\fontsize{12}\selectfont\text{$n(SUE)'.$}[/tex]
[tex]\noindent\rule{12cm}{0.4pt}[/tex]
b) Please have a look at the image I posted for the Venn diagram.
[tex]\noindent\rule{12cm}{0.4pt}[/tex]
c) Find the number of students who like to study economics.
[tex]\fontsize{12}\selectfont\text{$\bullet$\ Total number of students, $n(U)=100.$}[/tex]
[tex]\fontsize{12}\selectfont\text{$\bullet$ Students who like only environmental science, $n_o(S)=30$}[/tex]
[tex]\fontsize{12}\selectfont\text{$\bullet$\ Students who like only economics, $n_o(E)=40.$}[/tex]
[tex]\fontsize{12}\selectfont\text{$\bullet$\ Students who don't like any subjects, $n(SUE)'=10$}[/tex]
Using the formula:
[tex]\fontsize{12}\selectfont\text{$n(U)=n_o(S)+n_o(E)+n(S\cap E)+n(SUE)'$}[/tex]
Substitute the known values:
[tex]\fontsize{12}\selectfont\text{$100=30+40+n(S\cap E)+10$}[/tex]
Solve for [tex]\fontsize{12}\selectfont\text{$n(S\cap E):$}[/tex]
[tex]\fontsize{12}\selectfont\text{$100=80+n(S\cap E)$}[/tex]
[tex]\fontsize{12}\selectfont\text{$n(S\cap E)=20$}[/tex]
So, 20 students like both environmental science and economics.
To find the number of students who like to study economics, we consider:
[tex]\fontsize{12}\selectfont\text{$n(E)=n_o(E)+n(S\cap E)$}[/tex]
Substitute the known values:
[tex]\fontsize{12}\selectfont\text{$n(E)=40+20=60$}[/tex]
[tex]\noindent\rule{12cm}{0.4pt}[/tex]
d) Is n(E) = 30% of the total students? Give reason.
- We found that the number of students who like to study economics is 60.
- The total number of students in the class is 100.
[tex]\fontsize{12}\selectfont\text{To find if $n(E)$ is 30\% of the total students, we calculate:}[/tex]
[tex]\fontsize{12}\selectfont\text{$\dfrac{n(E)}{n(U)}\times100\%.$}[/tex]
[tex]\fontsize{12}\selectfont\text{Substituting the values, we get:$}[/tex]
[tex]\fontsize{12}\selectfont\text{$\dfrac{60}{100}\times100\%=60\%$}[/tex]
Therefore, n(E) is not 30% of the total students.[tex]\noindent\rule{12cm}{0.4pt}[/tex]
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