Out of 30 students in a hostel. 15 study History and 8 study Economics and 5M 6 study Geography. It is known that 3 students study all these subjects. Show that 7 or more students study none of these subjects.
Answers
Answer:
give questions clearly!! can't understand
The correct question is,
Out of 30 students in a hostel, 15 study History, 8 study Economics, and 6 study Geography. It is known that 3 students study all these subjects. Show that 7 or more students study none of these subjects.
Ans: |H∪E∪G|′≥7.
Given:
Out of 30 students in a hostel. 15 study History and 8 study Economics and 5M 6 study Geography. It is known that 3 students study all these subjects.
To find:
Show that 7 or more students study none of these subjects
Solution:
Here, they gave n(H)=15, n(E)=8, n(G)=6, n(H∩E∩G)=3. (For a set A, n(A) denotes how many members it has.)
⇒ |H∪E∪G|+|H∪E∪G|′=30
⇒ |H∪E∪G|=|H|+|E|+|G|−|H∩G|−|G∩E|−|E∩H|+|H∩E∩G|
⇒ |H∩G|+|G∩E|+|E∩H|=|H|+|E|+|G|+|H∩E∩G|−|H∪E∪G|
⇒ |H∩G|+|G∩E|+|E∩H|=23−|H∪E∪G|
But |H∩G|+|G∩E|+|E∩H|≥0, because are non-negative integers.
Then |H∪E∪G|≤23 this implies |H∪E∪G|′≥7.
Hence, |H∪E∪G|′≥7.
#SPJ2