In a class of 225 students, it was found that 110 students liked Mathematics, 115 students liked Science and 20 students liked none of the subjects.
a. If M and S denote the set of students who liked Mathematics and Science respectively then write the cardinality of n(MUS)
b. Represent the above information in a Venn-diagram.
c. Compute the number of students who liked both the subjects.
d. How many students liked at most one subjects?

Let's solve the given problem step-by-step:
a. Find the cardinality of n ( M ∪ S )
To find the number of students who liked either Mathematics or Science or both (denoted as n ( M ∪ S ) ), we can use the following formula from set theory:
n ( M ∪ S ) = n ( M ) + n ( S ) − n ( M ∩ S )
Where:

n ( M ) is the number of students who liked Mathematics, which is 110.
n ( S ) is the number of students who liked Science, which is 115.

We are initially not given n ( M ∩ S ) (students who liked both subjects), but we can calculate it later. Meanwhile, to find n ( M ∪ S ) , we consider that 20 students liked none of the subjects. Therefore, the students who liked at least one subject is:
n ( M ∪ S ) = 225 − 20 = 205
b. Venn Diagram Representation
A Venn diagram is a visual way to represent sets and their relationships. Here is how the sets would look:

Draw two overlapping circles, one labeled 'Mathematics' (M) and the other labeled 'Science' (S).
The intersection of the two circles represents the students who liked both Mathematics and Science.
Circle 'M' represents all students who liked Mathematics (110 students).
Circle 'S' represents all students who liked Science (115 students).
The area outside of these circles but within the rectangle represents the 20 students who liked none of the subjects.

c. Compute the number of students who liked both subjects
We use the principle of inclusion and exclusion:
205 = 110 + 115 − n ( M ∩ S )
Solving for n ( M ∩ S ) :
205 = 225 − n ( M ∩ S ) 225 − 205 = n ( M ∩ S ) n ( M ∩ S ) = 20
So, 20 students liked both Mathematics and Science.
d. How many students liked at most one subject?
To find the number of students who liked at most one subject, we consider those who liked only one subject or none:
n ( M ∖ S ) + n ( S ∖ M ) + n ( N o n e )
Where:

n ( M ∖ S ) are the students who liked only Mathematics: n ( M ∖ S ) = n ( M ) − n ( M ∩ S ) = 110 − 20 = 90
n ( S ∖ M ) are the students who liked only Science: n ( S ∖ M ) = n ( S ) − n ( M ∩ S ) = 115 − 20 = 95

Adding them up including those who liked none (20 students):
90 + 95 + 20 = 205
Thus, there are 185 students who liked at most one subject.
This comprehensively answers all parts of the question using set theory principles.

Answered by EmmaGraceJohnson | 2025-07-07