sets

4. a) A survey was conducted among 125 students of class 10 studying in Gyan Mandir Secondary School regarding the places for educational tour and it was found that 65 students preferred to visit Lumbini (L), 75 preferred to visit Pokhara (P), and 25 preferred both the places.

1. Write the cardinality of set of students who preferred to visit both Pokhara and Lumbini in set notational form.
2. Show the above information in a Venn-diagram.
3. Find the number of students who preferred neither of two places.
4. Find the ratio of the number of students who preferred Lumbini only and Pokhara only.

📝 Given Information

Total students, \(n(U) = 125\)

Preferred Lumbini, \(n(L) = 65\)

Preferred Pokhara, \(n(P) = 75\)

Preferred both, \(n(P \cap L) = 25\)

(a) Write the cardinality of set of students who preferred to visit both Pokhara and Lumbini in set notational form.

The cardinality of the set of students who preferred both places is denoted by the intersection of sets P and L.

\(n(P \cap L) = 25\)

(b) Show the above information in a Venn-diagram.

(c) Find the number of students who preferred neither of two places.

First, let's find the number of students who preferred at least one place (Union):
\(n(P \cup L) = n(P) + n(L) - n(P \cap L)\)
\(n(P \cup L) = 75 + 65 - 25\)
\(n(P \cup L) = 115\)

The number of students who preferred neither is the complement of the union:
\(n(\overline{P \cup L}) = n(U) - n(P \cup L)\)
\(n(\overline{P \cup L}) = 125 - 115 = 10\)

\(n(\overline{P \cup L}) = 10\)

(d) Find the ratio of the number of students who preferred Lumbini only and Pokhara only.

Number of students who preferred Lumbini only:
\(n_0(L) = n(L) - n(P \cap L)\)
\(n_0(L) = 65 - 25 = 40\)

Number of students who preferred Pokhara only:
\(n_0(P) = n(P) - n(P \cap L)\)
\(n_0(P) = 75 - 25 = 50\)

Ratio = \(n_0(L) : n_0(P)\)
Ratio = \(40 : 50\)

Ratio = 4 : 5