Sets: Finding the minimum and maximum values of n(A n B) and n(A U B)'

[dkrbabajee]

Questions on finding the minimum and maximum values of n(A n B) and n(A U B)'
can be answered by considering 3 cases.
Case 1:
A C B, A is a subset of B,

Case 2:
E = A U B, there is nothing outside A and B, E is the universal set,

Case 3:
A and B are disjoint sets.


Qu1:
n(E)=50, n(A)=26 and n(B)=34,
Solution:
Case 1:
Since A C B, we have n(A n B)=n(A)=26 and n(A U B)'=n(E)-n(B)=50-34=16,

Case 2:
Obviously, n(A U B)'=0, 
n(A n B)=n(A)+n(B)-n(E)=34+26-50=10,

Case 3:
Obviously, n(A n B)=0,
n(A)+n(B)=34+26=60>n(E),
A and B cannot be disjoint.
Case 3 is not possible.

From Cases 1 and 2,
min n(A n B)=10
max n(A n B)=26
min n(A U B)'=0
max n(A U B)'=16


Qu2:
n(E)=50, n(A)=20 and n(B)=25,
Solution:
Case 1:
Since A C B, we have n(A n B)=n(A)=20 and n(A U B)'=n(E)-n(B)=50-25=25,

Case 2:
Obviously, n(A U B)'=0, 
n(A)+n(B)=20+25=45<n(E),
There must be some elements outside A and B.
Case 2 is not possible.


Case 3:
Obviously, n(A n B)=0,
n(A U B)'=n(E)-(n(A)+n(B))=50-(20+25)=5,


From Cases 1 and 3,
min n(A n B)=20
max n(A n B)=0
min n(A U B)'=5
max n(A U B)'=25

Comments:
Case 1 is always possible.
You can draw the Venn diagram for the 3 cases of each question.

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