Set Theory Calculation for Player Groups
This problem involves finding the number of players in the intersection of two sets (Hockey and Cricket) within a larger group, given information about the total number of players and the sizes of various overlapping sets.
Applying Inclusion-Exclusion Principle
We use the Principle of Inclusion-Exclusion for three sets: H (Hockey), F (Football), and C (Cricket).
The formula is:
$|H \cup F \cup C| = |H| + |F| + |C| - |H \cap F| - |H \cap C| - |F \cap C| + |H \cap F \cap C|$Given values:
- Total players, $|H \cup F \cup C| = 44$
- Hockey players, $|H| = 26$
- Football players, $|F| = 24$
- Cricket players, $|C| = 24$
- Hockey and Football, $|H \cap F| = 8$
- Football and Cricket, $|F \cap C| = 12$
- All three games, $|H \cap F \cap C| = 5$
We need to find the number of players who play both Hockey and Cricket, denoted as $|H \cap C|$.
Solving for Hockey and Cricket Intersection
Substitute the known values into the Inclusion-Exclusion formula:
$44 = 26 + 24 + 24 - 8 - |H \cap C| - 12 + 5$Combine the known numbers:
$44 = (26 + 24 + 24 + 5) - (8 + 12) - |H \cap C|$ $44 = 79 - 20 - |H \cap C|$ $44 = 59 - |H \cap C|$Now, isolate $|H \cap C|$:
$|H \cap C| = 59 - 44$ $|H \cap C| = 15$Final Answer
Therefore, 15 players play both hockey and 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 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 : - Suppose three meetings of a group of professors were arranged in Mumbai, Delhi and Chennai. Each professor of the group attended exactly two meetings. 21 professors attended Mumbai meeting, 27 attended Delhi meeting and 30 attended Chennai meeting. How many of them attended both the Chennai and Delhi meetings?
