Solving for Students Who Like At Least One Drink
This problem involves finding the total number of students who like cold drinks, milkshakes, or both, based on survey data. We can use the Principle of Inclusion-Exclusion.
Defining Sets and Given Values
- Let $C$ be the set of students who like cold drinks.
- Let $M$ be the set of students who like milkshakes.
- Number of students who like cold drinks: $|C| = 140$.
- Number of students who like milkshakes: $|M| = 120$.
- Number of students who like both cold drinks and milkshakes: $|C \cap M| = 80$.
- Total students surveyed: 200 (This information is not directly needed for calculating the union but provides context).
Applying the Inclusion-Exclusion Principle
We need to find the number of students who like at least one of the drinks, which is represented by the union of the two sets, $|C \cup M|$. The formula is:
$ |C \cup M| = |C| + |M| - |C \cap M| $Calculation
Substitute the given values into the formula:
$ |C \cup M| = 140 + 120 - 80 $ $ |C \cup M| = 260 - 80 $ $ |C \cup M| = 180 $Conclusion
Therefore, 180 students like at least one of the drinks (cold drinks or milkshakes).
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 :
