Set Theory Problem

Here are the solutions to the questions:

1. Set Theory Problem
Given:
Total students, $n(U) = 120$
Students who liked cricket, $n(C) = 60$
Students who liked basketball, $n(B) = 55$
Students who did not like any of the two games, $n(C \cup B)' = 20$

a) If C and B denote the sets of students who liked cricket and basketball game respectively, write the cardinality of $n(B \cup C)$.
We know that $n(U) = n(C \cup B) + n(C \cup B)'$.

$$120 = n(C \cup B) + 20$$ $$n(C \cup B) = 120 - 20$$ $$n(C \cup B) = 100$$

The cardinality of $n(B \cup C)$ is $\boxed{100}$.

b) Present the above information in a Venn-diagram.
First, find the number of students who liked both games, $n(C \cap B)$.
We use the formula: $n(C \cup B) = n(C) + n(B) - n(C \cap B)$.

$$100 = 60 + 55 - n(C \cap B)$$ $$100 = 115 - n(C \cap B)$$ $$n(C \cap B) = 115 - 100$$ $$n(C \cap B) = 15$$

Now, find the number of students who liked cricket only, $n_0(C)$.

$$n_0(C) = n(C) - n(C \cap B) = 60 - 15 = 45$$

And the number of students who liked basketball only, $n_0(B)$.

$$n_0(B) = n(B) - n(C \cap B) = 55 - 15 = 40$$

The Venn diagram is as follows:
(Imagine two overlapping circles. The left circle represents Cricket (C), the right circle represents Basketball (B). The overlapping region is $C \cap B$. The rectangle enclosing both circles represents the Universal set (U).)

• In the region for C only: 45
• In the region for B only: 40
• In the intersection region ($C \cap B$): 15
• Outside both circles (neither liked): 20

c) Find the number of students who liked cricket only.
The number of students who liked cricket only is $n_0(C)$.

$$n_0(C) = n(C) - n(C \cap B) = 60 - 15 = 45$$

The number of students who liked cricket only is $\boxed{45}$.

d) Compare the number of students who liked both cricket and basketball and who liked except these two games.
Number of students who liked both cricket and basketball, $n(C \cap B) = 15$.
Number of students who liked neither of the two games, $n(C \cup B)' = 20$.
Comparing these two values: $20 > 15$.
Therefore, the number of students who liked neither game is greater than the number of students who liked both games.

2. Simple and Compound Interest Problem
Given:
Principal, $P = \text{Rs. } 60,000$
Time, $T = 2$ years
Rate, $R = 10\%$ p.a.

a) Define compound interest.
Compound interest is the interest calculated on the initial principal, which also includes all of the accumulated interest from previous periods on a deposit or loan. It is often referred to as "interest on interest."

b) Find the compound interest received by Bijaya after 2 years.

Step 1: Calculate the amount ($A$) using the compound interest formula.