There are 25 students in a class. Of these, 14 speak Spanish, 12 speak French, 6 speak French and Spanish, 5 speak German and Spanish, 2 speak all three languages. Each of the 6 who speak German speaks another one of these languages as well.
(a) How many speak both French and German?
(b) How many speak none of these three languages?
Ive tried drawing a venn diagram and then using set operation (eg A/B ∪ C) but I cant seem to properly do it. Draw the diagram, that is.
Any help appreciated. Thanks
EDIT: I figured I could improve the answer by adding set notation too, so I have done this after each paragraph.
$2$ people speak all three languages, $0$ people speak German and nothing else, and $5 $ people speak German and Spanish. From this, and from drawing a Venn diagram, we see $3$ people speak Spanish and German only.
To write this with set notation, $$\begin{align}|G\cap S\cap(\neg F)|&=|G\cap S|-|G\cap S\cap F|\\&=5-2\\&=3.\end{align}$$
$6$ people speak German in total, $5$ have been accounted for above, and so $1$ person speaks just German and French, and so in total, $1+2=3$ people speak French and German. Part (a) answered.
To write this with set notation, $$\begin{align}|G\cap F|&=|G|-|G\cap(\neg F)|\\&=|G|-|G\cap S\cap(\neg F)|-|G\cap(\neg S)\cap(\neg F)|\\&=6-3-0\\&=0.\end{align}$$
$6$ people speak French and Spanish, of which $2$ also speak German and so $4$ people speak French and Spanish only. $12$ people speak French, of which $6$ speak Spanish too, and $3$ speak German too. So we would think this means $9$ people speak another language as well as French. However inclusion exclusion says that we counted the speakers of all three languages twice, so we subtract $2$. Thus $12-(9-2)=5$ people speak French only. Similar thought processes for Spanish only gives $5$ there too, and summing all the Venn entries gives $20$. So $25-20=5$ people speak none of the languages. Part (b) answered.
To write this with set notation, $$\begin{align}|F\cap S\cap(\neg G)|&=|F\cap S|-|F\cap S\cap G|\\&=6-2\\&=4\\|F\cap(\neg S)\cap(\neg G)|&=|F|-|F\cap G|-|F\cap S|+|F\cap G\cap S|\\&=12-|F\cap G\cap S|-|F\cap G\cap(\neg S)|-6+|F\cap G\cap S|\\&=6-|F\cap G\cap(\neg S)|\\&=6-(|G|-|G\cap S|-|G\cap(\neg S)\cap(\neg F)|)\\&=6-(6-5-0)\\&=5\\\text{Similar for Spanish only}&\implies|S\cap(\neg F)\cap(\neg G)=5\\|F\cup G\cup S|&=|F|+|G|+|S|-|F\cap G|-|F\cap S|-|S\cap G|+|F\cap G\cap S|\\&=12+6+14-3-6-5+2\\&=20\\\implies|\neg F\cap\neg G\cap\neg S|&=|\Omega|-|G\cup S\cup F|\text{ where }\Omega\text{ is the entire set}\\&=25-20\\&=5\end{align}$$