Set Theory Worked Examples for IGCSE Maths

Question

In a group of 40 students, 25 study French (F), 20 study Spanish (S), and 10 study both. Complete the Venn diagram and find n(F union S)'.

Solution

1. 1

   Fill in the intersection

   F intersection S = 10

   Start with the overlap.

2. 2

   Fill in the remaining parts

   F only = 25-10 = 15. S only = 20-10 = 10.

   Subtract the intersection from each set total.

3. 3

   Find those outside both sets

   Outside = 40 - 15 - 10 - 10 = 5

   Total minus all regions inside the circles.

4. 4

   Find (F union S)'

   (F union S)' = 5

   The complement of the union is everything outside both circles.

Question

100 people were surveyed about sports. 50 play football (F), 35 play cricket (C), 30 play tennis (T). 15 play F and C, 10 play F and T, 12 play C and T, 5 play all three. Complete the Venn diagram.

Solution

1. 1

   Fill in the centre

   F intersection C intersection T = 5

   Always start from the innermost region.

2. 2

   Fill in the two-set intersections

   F and C only = 15-5 = 10. F and T only = 10-5 = 5. C and T only = 12-5 = 7.

   Subtract the triple intersection from each pair.

3. 3

   Fill in the single-set regions

   F only = 50-10-5-5 = 30. C only = 35-10-7-5 = 13. T only = 30-5-7-5 = 13.

   Subtract all overlapping regions from each set total.

4. 4

   Find outside region

   Outside = 100 - 30 - 13 - 13 - 10 - 5 - 7 - 5 = 17

   Total minus all seven regions inside the circles.

Question

From the Venn diagram above, find: (a) n(F intersection C'), (b) P(a person plays exactly one sport).

Solution

1. 1

   Find F intersection C'

   F intersection C' means in F but NOT in C. That is F only + (F and T only) = 30 + 5 = 35.

   C' means not in C. So we want the parts of F that do not overlap with C.

2. 2

   Find P(exactly one sport)

   Exactly one = F only + C only + T only = 30 + 13 + 13 = 56. P = 56/100 = 0.56.

   Exactly one means in one set but not in any intersection.