How do I determine the cardinality of the following sets given inclusion-exclusion principle?
- The number of people in a test group of 100 people who do not speak English, French, or Russian. The following is known: 48 people speak Russian, 45 people can speak French and 67 people speak English. Exactly 13 of the people speak all 3 languages. In addition, 29 people can speak English and French, 27 people speak English and Russian and 18 people speak Russian and French.
So, I know that the sets here are not disjoint, which is why we’re using the inclusion-exclusion principle.
For 1), I did the following:
a. |testGroup| = |TG| == 100 (who don’t speak English, French, or Russian).
b. |englishSpeakers| = |E| == 67;
c. |frenchSpeakers| = |F| == 45;
d. |russianSpeakers| = |R| == 48;
e. |E ∩ F ∩ R | = 13;
f. |E ∩ F| = 29;
g. |E ∩ R| = 27;
h. |F ∩ R| = 18;
To find “pure” englishSpeakers: |EP| = |E| - ( |E ∩ F| U |E ∩ R| U |E ∩ F ∩ R | ) = 7.
To find “pure” frenchSpeakers: |FP| = |F| - ( |E ∩ F| U |E ∩ F ∩ R | ) = 3.
To find “pure” russianSpeakers: |RP| = |R| - (|E ∩ R| U |E ∩ F ∩ R | ) = 17.
Here, I imagined the sets E, F, and R to contain subsets of points e, f, and g. In other words, sets E, F, and R are actually intersections between one another, including |EP|, |FP|, |RP|.
It seems that |TG| is actually the complement of |E| U |F| U |R|. If that’s the case, I'd have to find it, but I'm still figuring that out (assuming the hypothesis holds).
However, I’m not sure if this is the correct path to go about it. I have a masters in Management and now I’m pursuing a bachelor in computer science, because I was always interested in it, given that it helped solve business problems fairly easily (or, at least, made certain problems easier to solve).
I see the value in combinatorics (and discrete mathematics in general), because I can actually use it for things like describing and predicting customer demand, as well as improving negotiation processes and outcomes. (I have done so in the past, and it helped).
Because I don’t have a background in discrete mathematics, I’d be happy if you could share with me both the solutions and the rationale behind it, as it would greatly help me understand the principles behind it, and the problem-solving process.
Thanks!
The Inclusion-Exclusion Principle for three sets states that $$|E \cup F \cup R| = |E| + |F| + |R| - |E \cap F| - |E \cap R| - |F \cap R| + |E \cap F \cap R|$$ The number who speak none of the languages is found by subtracting $|E \cup F \cup R|$ from the number of people in the test group.
If you want to find the number of people who speak only English, you have to subtract the number of people who speak English and another language from the number of people who speak English. \begin{align*} |E \cap F' \cap R'| & = |E| - |E \cap (F \cup R)|\\ & = |E| - |(E \cap F) \cup (E \cap R)|\\ & = |E| - (|E \cap F| + |E \cap R| - |E \cap F \cap R|)\\ & = |E| - |E \cap F| - |E \cap R| + |E \cap F \cap R| \end{align*}