Analyzing the Survey Data
The goal is to find the number of people who like neither tea nor coffee from the given survey results.
Given Survey Information
- Total surveyed: 100
- Likes Tea (|T|): 60
- Likes Coffee (|C|): 50
- Likes Both Tea and Coffee (|T ∩ C|): 25
Calculating Union of Preferences
First, determine the number of people who like at least one beverage (Tea or Coffee or Both). This is calculated using the Principle of Inclusion-Exclusion:
$|T \cup C| = |T| + |C| - |T \cap C|$
Substituting the values:
$|T \cup C| = 60 + 50 - 25$
$|T \cup C| = 110 - 25$
$|T \cup C| = 85$
So, 85 people like tea, coffee, or both.
Determining Those Who Like Neither
To find the number of people who like neither drink, subtract the count of those who like at least one drink from the total number of people surveyed:
Number liking Neither = Total People - $|T \cup C|$
Calculation:
$100 - 85 = 15$
Therefore, 15 people like neither tea nor coffee.
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 :
