Question

Set theory

Answer 1

In the stafford public school , students had an option to study none or
one or more of three foreign languages viz: french, spanish and
german. The total student strength of school was \(2116\) students out
of which \(1320\) students studied french and \(408\) students studied
both french and spanish. The number of people who studied german
was found to be \(180\) higher than the number of students who studied
spanish. It was also observed that \(108\) students studied all three
subjects.

What is the maximim possible number of students who didn't studied
any of the three languages ?
\(a.)\ 890 \ \ \ b.) 796 \ \ \ c.)\ 720 \ \ \ d.)\ \text{none of these} \)

What is the minimum posibble number of studnets who didn't study
any of three languages ?
\(a.)\ 316 \ \ \ b.)\ 0 \ \ \ c.)\ 158 \ \ \ d.)\ \text{none of these} \)

if the number of students who used to speak only french was \(1\)
more than the number of people who used to speak only german ,
then whay could be the maximum number of people who used to speak
only spanish ?
\(a.)\ 413 \ \ \ b.)\ 398 \ \ \ c.)\ 403 \ \ \ d.)\ 431 \)

Answer 2

\(\large \color{black}{\begin{align}
& U=(f\cup s\cup g)+x \hspace{.33em}\\~\\
& 2116=(f\cup s\cup g)+x \hspace{.33em}\\~\\
& (f\cup s\cup g)=f+g+s-(f\cap g)-(f\cap s) -(s\cap g)+(s\cap g \cap f) \text{}\hspace{.33em}\\~\\


& 2116-x=1320+s+180+s-f\cap g-408 -s\cap g+108 \text{}\hspace{.33em}\\~\\

\end{align}}\)

Answer 3

\(\large \color{black}{\begin{align}



&x= s\cap g-916-2s+f\cap g \text{}\hspace{.33em}\\~\\

\end{align}}\)

Answer 4

@phi

Answer 5

this one is painful. Based on all the info, I think we can let the german and spanish both be zero except for where they intersect with french. i.e. everybody takes french
that would leave 2116-1320 for the number who take no language

Answer 6

u mean these are \(0\)

Answer 7

or are u saying these to be \(0\)

Answer 8

and also german & spanish region = 0

Answer 9

ok

Answer 10

then the # taking spanish is 408
the # taking german is 588 split into 108 and 480

Answer 11

We want to minimize the number of people in the circles, but we know the french circle has 1320. That leaves us to put 0 everywhere we can in the other languages, but still match the criteria. I believe we can do that as shown above.

the # not taking a language is then the 2116-1320

Answer 12

yea i got that logic of 1st question

Answer 13

For the min not taking a language , we want to maximize the # in the circles. To do that we want to minimize the # in the "overlap" regions

Answer 14

so we want these to be \(0\) now

Answer 15

yes

Answer 16

wouldn't the answer be \(0\)

Answer 17

yes, I think so, but it would be good to demonstrate an assignment