Number of Elements in named Subsets

When determining the number of elements in named subsets of two intersecting sets, given the number of elements in some of the other subsets, it is wise to firstly:

- Decide what letters you will use to represent the subsets.
- List the given information of elements.
- Use a Venn diagram and or formulae to solve for the elements in question.

Example 1
In a class of 35 students, 25 studied Mathematics, 30 studied English Language and 20 students studied both Mathematics and English language. Determine the number of students who studied:
(a) Mathematics only
(b) English Language only.

Let, M = {students who studied mathematics}
E = {students who studied English language)
Given information:
n(M) = 25
n(E) = 30
n(M ∩ E) = 20

From the given information, it can be deduced that,
n(U) = n(M ⋃ E) = 35

Using a Venn diagram:

Using Formulae:

(a) Number of students who studied Mathematics only
n(M ∩ E’) / n(M only) = n(M) – n(M ∩ E)
= 25 – 20
= 5

(b)Number of students who studied English language only
n(M’ ∩ E) / n(E only) = n(E) – n(M ∩ E)
= 30 – 20
= 10

Example 2
At a youth club of 40 members, 25 like football, 20 like cricket and 5 like neither football nor cricket. Determine the number of members who like:
(a) both football and cricket
(b) football only
(c) cricket only.

Let, F = {members who like football}
C = {members who like cricket}
Given information:
n(U) = 40
n(F) = 25
n(C) = 20
n(F ⋃ C)’ = 5

Let x represent the number of members who like both football and cricket, that is,
n(F ∩ C) = x

Then, the number of members who like football only is represented by,
n(F ∩ C’) = 25 –x

Also, the number of members who like cricket only is represented by,
n(C ∩ F’ ) = 20 –x

Using a Venn diagram:

Solving:

(a) Members who like both football and cricket
n(U) = 25 – x + x + 20 – x + 5
since n(U) = 40, then
40 = 25 – x + x + 20 – x + 5
40 = 25 + 20 + 5 –x + x – x
40 = 50 – x
x = 50 – 40
x = 10
that is, x= 10 members

(b) Members who like football only
n(F ∩ C’) = 25 –x members
= 25 – 10
= 15

(c) Members who like cricket only
n(F’ ∩ C) = 20 –x members
= 20 – 10
= 10

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