In a group of 15 people, 7 read French, 8 read English while 3 of them read none of these two. How many of them read French and English both:
Explanation
To solve this problem clearly, we can use basic set theory principles or a conceptual breakdown of the groups.
1. Identifying the Universal Set Component
Let the total number of people in the entire group be represented as the Universal set, n(U).
n(U)=15
The problem states that 3 people do not read either of the two languages. Therefore, we find the number of people who read at least one language (the union of the two sets, n(F∪E)) by subtracting the non-readers from the total group size:
n(F∪E)=Total People−People who read none
n(F∪E)=15−3=12
This means exactly 12 people read either French, English, or both languages.
2. Setting Up the Set Theory Formula
Let us define our individual language counts:
Number of French readers, n(F)=7
Number of English readers, n(E)=8
We need to find the number of people who read both languages, which is represented as the intersection of the two sets, n(F∩E).
According to the fundamental formula of set theory:
n(F∪E)=n(F)+n(E)−n(F∩E)
3. Calculating the Intersection Value
Substitute the known values into our formula:
12=7+8−n(F∩E)
12=15−n(F∩E)
Now, isolate n(F∩E) by rearranging the equation:
n(F∩E)=15−12
n(F∩E)=3