Question 1151779
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A = set of people who study Arabic
F = set of people who study French
Y = set of people who study Yoruba
U = universal set = set of everyone surveyed (100 people total)


Draw a Venn Diagram of 3 circles representing the three sets of language classes.
<img width = "35%" src = "https://i.imgur.com/epMLn05.png">
Take note of the 8 distinct regions which I've marked in blue
Region (a) = set of people taking Arabic only
Region (b) = set of people taking Arabic and French only
Region (c) = set of people taking French only
Region (d) = set of people taking Arabic and Yoruba only
Region (e) = set of people taking all three languages
Region (f) = set of people taking French and Yoruba only
Region (g) = set of people taking Yoruba only
Region (h) = set of people taking neither of the three classes mentioned


We have 3 people studying all three languages, so that goes in the very center region (e).


There are 10 people taking Arabic and Yoruba, so that leaves 10-3 = 7 who are taking Arabic and Yoruba only (and not French). This goes in region (d)


There are 8 people taking Arabic and French, leaving 8-3 = 5 people taking these two languages only (not Yoruba). This goes in region (b).


Then we have a '2' in region (f) because there are 5 people studying French and Yoruba, so 5-3 = 2 are just studying these two languages (not Arabic).



We have this partially filled out Venn Diagram so far
<img width = "35%" src = "https://i.imgur.com/6WsUhwZ.png">
We know that 28 students are studying Arabic. So far we have 5, 3, and 7 in regions (b), (e) and (d). The total so far is 5+3+7 = 15. That leaves 28-15 = 13 who study Arabic only (and not French nor Yoruba). Write "13" in region (a).


Then there are 30 French students. Subtract off the values that are in circle F, and we have 30-5-3-2 = 20 students who are studying French only (neither Arabic nor Yoruba). Write this in region (c).


Next up there are 42 students in a Yoruba language class. Subtract off the values in circle Y: 42-7-3-2 = 30. Write "30" in region (g) to indicate we have 30 students who are only taking this language class (and neither French nor Arabic)


Lastly, we add all of the values we have filled out so far in regions (a) through (g)
13+5+20+7+3+2+30 = 80
There are 80 students who are taking at least one language class mentioned here (ie they are taking one or more of the three classes)


However, there were 100 people surveyed as part of the universal set. So we have 100-80 = 20 people taking neither of the three language courses. Write this in region (h)



This is the fully completed Venn Diagram
<img width = "35%" src = "https://i.imgur.com/BNPVxLJ.png">
The region labels are optional, but I find them helpful to be able to describe the diagram values.


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Once you have the Venn Diagram, the questions are fairly straight forward


Question (1). How many student where studying no language
Answer: 20 students (see region (h))


Question (2). How many student had Yoruba as their only language
Answer: 30 students (see region (g))


Question (3). How many student study French if and only if study Yoruba
Answer: 5 students (combine regions (e) and (f))
I interpret this question to be asking "how many students are studying Yoruba and French at the same time?". This includes the 3 people taking all three languages.
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