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)
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?
