Answer:
20 people
Explanation:
Let A be the set of people who liked modern songs, and B be the set of people who liked folk songs. Let x be the number of people who liked both modern and folk songs, and y be the number of people who liked modern songs but not folk songs.
We are given the following information:
|A| = 40 (40 people liked modern songs)
|B| = y + x + 30 (30% of the group liked modern songs but not folk songs, which means that the remaining 70% liked folk songs either exclusively or in combination with modern songs. So, |B| = 0.7 * 80 = 56, and we can express this in terms of x and y as y + x + 30 = 56.)
|B\A| = 30 (30 people liked folk songs only)
We want to find the number of people who did not like either modern or folk songs, which is the complement of the set (A U B). We can use the formula for the size of the union of two sets:
|A U B| = |A| + |B| - |A intersect B|
Substituting the values we have, we get:
80 - |A U B| = 40 + (y + x + 30) - x
Simplifying, we get:
80 - |A U B| = 70 + y + 30
Simplifying further, we get:
|A U B| = 40 - y - x
We also know that:
|B\A| = 30
Substituting the formula for the set difference, we get:
|B| - |A intersect B| = 30
Substituting the values we have, we get:
y + x + 30 - x = 30
Simplifying, we get:
y = 0
This means that no one liked modern songs but not folk songs.
Substituting y = 0 into the equation for |A U B|, we get:
|A U B| = 40 - x
Substituting this into the equation we derived earlier, we get:
80 - (40 - x) = 70 + 0 + 30
Simplifying, we get:
x = 20
So, the answer is 20 people.