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)
Match List-I with List-II
List-1 List-II (A) If X and Y are two sets such that n(X)= 17, n(Y)=23, n(X $\cup$ Y)=38, then n(X $\cap$ Y) is (I) 20 (B)) If n(X) = 28,n(Y) = 32,n(X$\cap$Y) = 10, then n(X$\cup$Y) is (II) 10 (C) If n(X) = 10, then n(7X) is (III) 50 (D) If n(Y) = 20, then n($\frac{Y}{2}$) is (IV) 2 Choose the Correct answer from the options given below:
- From the given sets, which is an infinite set:
1. {x: x $\in$ N and (x-1)(x-2) = 0}
2. {x: x $\in$ N and x is prime number and less than 199}
3. {x: x $\in$ N and x$^5$ - 1 = 0}
4. {x: x $\in$ N and x is odd} Consider the following relation R={(4,5),(5,4), (7,6),(6,7)} on set I={4,5,6,7}. Which of the following properties relation R does not have?
A. Reflexive property
B. Symmetric property
C. Transitive property
D. Antisymmetric property
Choose the correct answer from the options given below:Find the least upper bound and greatest lower bound of $S=\{X,Y,Z\}$ if they exist, of the poset whose Hasse diagram is shown below:
- In a class, 40 like Math, 35 like Science, 30 like English; 15 like both Math and Science, 12 like Math and English, 10 like Science and English, 5 like all three. How many like atleast one?
