Understanding the Problem
We need to find the number of people who do not play either volleyball or cricket out of a total of 2000 people. We are given the counts for volleyball players, cricket players, and those who play both.
Applying Set Theory Principles
Let V be the set of people who play volleyball and C be the set of people who play cricket.
- Total number of people = 2000
- Number of volleyball players, $|\text{V}| = 300$
- Number of cricket players, $|\text{C}| = 500$
- Number of people playing both, $|\text{V} \cap \text{C}| = 200$
Calculating Players of At Least One Game
We use the Principle of Inclusion-Exclusion to find the number of people who play at least one game (volleyball or cricket or both).
The formula is: $|\text{V} \cup \text{C}| = |\text{V}| + |\text{C}| - |\text{V} \cap \text{C}|$
Substituting the values:
$|\text{V} \cup \text{C}| = 300 + 500 - 200$
$|\text{V} \cup \text{C}| = 800 - 200$
$|\text{V} \cup \text{C}| = 600$
So, 600 people play at least one of the games.
Calculating People Playing Neither Game
To find the number of people who play neither game, subtract the number of people playing at least one game from the total number of people.
Number playing neither = Total people - $|\text{V} \cup \text{C}|$
Number playing neither = $2000 - 600$
Number playing neither = $1400$
Conclusion
Therefore, 1400 people do not play either volleyball or cricket.
Important Questions from Set Theory and Venn Diagram (Notes)
A student is free to choose only Chemistry, only Biology or both. If out of $32$ students, Chemistry has been chosen by $16$ and Biology by $25$, then how many students have chosen Biology but not Chemistry?
- Fourteen of the students in a class are girls. Eight students in the class wear blue shirts. Two are neither girls nor wear blue shirts. Five students who wear blue shirts are girls. How many students are there in the class?
- In a group of 44 players, 26 play hockey, 24 play football and 24 play cricket. Eight of them play both hockey and football, 12 play both football and cricket, and 5 play all the three games. How many play both hockey and cricket?
- In a group of students, 30% play only cricket, 20% play only football and 10% play only basketball. 20% of the students play both football and cricket, 15% play both basketball and cricket, 10% play both football and basketball. 15 students play no games, while 5% of the students play all three games. What is the total number of students?
- Which of the following statements are true ?
A. Set A = {x : x $\in$ R and 2 < x < 3} is a null set.
B. Set A = {x : x $\in$ R and 2 < x < 4} is a singleton set.
C. Set A = {x : x $\in$ R and 1 < x < 9} is an infinite set.
D. Set A = {x : x $\in$ R and 1 < x < 9} is a finite set.
E. Set A = {a, b, c, d, e} and B = {c, d, a, e, b} are equal sets.
Choose the correct answer from the options given below :
