Set Theory Application: Identifying Groups
This problem involves classifying students based on two criteria: gender (girl or not) and shirt color (blue shirt or not). We are given information about overlapping groups.
- Number of girls (G): 14
- Number of students wearing blue shirts (B): 8
- Number of girls wearing blue shirts (G and B): 5
- Number of students who are neither girls nor wear blue shirts: 2
Calculating Union of Groups (G or B)
We use the principle of inclusion-exclusion to find the number of students who are either girls OR wear blue shirts (or both). The formula is:
$ |G \cup B| = |G| + |B| - |G \cap B| $
Substituting the given values:
$ |G \cup B| = 14 + 8 - 5 $
$ |G \cup B| = 22 - 5 $
$ |G \cup B| = 17 $
So, 17 students are either girls or wear blue shirts or both.
Final Total Calculation
The total number of students in the class includes those who are girls or wear blue shirts (or both), plus those who are neither.
Total Students = (Students in G or B) + (Students neither G nor B)
Total Students = $ |G \cup B| + 2 $
Total Students = $ 17 + 2 $
Total Students = 19
Therefore, there are 19 students in the class.
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?
- 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 : - 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?
