Interpreting 3 Circled Venn Diagrams

Question

The Venn diagram above represents the 20 students who took one or more of the 3 available art classes or took no art classes at all. two students took no classes at all. 1.How many people take only two classes?

2. How many elements are there in the intersection of Photography and Sculpture? Why is the number of these elements 3 instead of 2? If the number of elements were 3, then wouldn't we be including the elements common to painting in addition to photography and sculpture?

3. I have difficulty with venn diagrams in general. Could you give me additional questions that will help me understand venn-diagrams? What are other questions that could be asked to interpret the diagram?

Answer 1

Consider the Venn diagram below.

The students who take Painting are represented by the red disk (including the regions shown in magenta, white, and yellow). The students who take Photography are represented by the blue disk (including the regions shown in magenta, white, and cyan). The students who take Sculpture are represented by the green disk (including the regions shown in yellow, white, and cyan). The students who do not take any of the art courses are represented by the light gray region inside the rectangle but outside the circles.

The students who take more than one course are represented by the regions in which the disks intersect.

1. The students who take exactly two courses are represented by the regions where exactly two of the disks intersect. There are two students in the magenta region who take Painting and Photography. There are three students in the yellow region who take Painting and Sculpture. There are two students in the cyan region who take Photography and Sculpture. Thus, the number of students taking exactly two of the courses is

$$2 + 3 + 2 = 7$$

2. The three students who take both Photography and Sculpture are represented by the cyan and white regions where the blue and green disks intersect. The two students who take only Photography and Sculpture are represented by the cyan region. The one student who also takes Painting is represented by the white region where all three disks intersect.

3. Other questions that could be asked include:

(a) How many students are enrolled in exactly one of the courses?

Answer: $10$

(b) How many students are enrolled in all three courses?

Answer: $1$

(c) How many students take both Painting and Photography?

Answer: $3$

(d) How many students take at least one of the courses Painting and Sculpture?

Answer: $15$