Question

If n(U) = 40 , n(A) = 25 and n(B) = 20 , Then Find:-
i) The greatest value of n ( A U B ) .
ii) The least value of n( A ∩ B )

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Answers :-
i) 40
ii) 5 ​

Answer 1

Given :-

n ( U ) = 40 , n ( A ) = 25 and n ( B ) = 20

To Find :-

i) The greatest value of n ( A U B ) .

ii) The least value of n( A ∩ B ) .

Used Concepts :-

For any sets A and B if the given sets are finite :-

• n ( A U B ) = n ( A ) + n ( B ) - n ( A ∩ B ) .
• n ( A ) and n ( B ) are the cardinal number/ Order of the respective sets .
• In any question related to sets if the universal set is given , then all sets are subsets of the universal set and all members of any set can't be from outside of the universal set.

Solution :-

i ) As n ( U ) is given i.e 40 , then cardinal number of any set can't be greater then that of the universal set

Therefore , n ( A U B ) ≤ n ( U )

=> n ( A U B ) ≤ 40

Hence , The greatest value of n ( A U B ) is 40.

ii) Here , n ( A U B ) ≤ 40

n ( A ) = 25

n ( B ) = 20

As we knows that ,

n ( A U B ) = n ( A ) + n ( B ) - n ( A ∩ B )

Putting the values , but here we didn't know the correct value of n ( A U B ) , So ,

n ( A ) + n ( B ) - n ( A ∩ B ) ≤ n ( A U B )

25 + 20 - n ( A ∩ B ) ≤ 40

45 - n ( A ∩ B ) ≤ 40

45 - 40 ≤ n ( A ∩ B )

5 ≤ n ( A ∩ B )

Therefore , least value of n ( A ∩ B ) is 5 .

Additional Information :-

For any sets A and B if provided that the sets are finite :-

• n ( A U B ) = n ( A ) + n ( B ) - n ( A ∩ B ) .
• n ( A U B ) = n ( A - B ) + n ( B - A ) + n ( A ∩ B ) .
• n ( A - B ) = n ( A U B ) - n ( B ) .
• n ( A - B ) = n ( A ) - n ( A ∩ B ) .
• n ( B - A ) = n ( A U B ) - n ( A ) .
• n ( B - A ) = n ( B ) - n ( A ∩ B ) .

Formula related to complement of a Set A :-

• If a set A is given and the universal set also , then complement of A is the set consisting all elements which didn't belong to A . Written as A^c .
• n ( A^c ) = n ( U ) - n ( A )
• ( ( A^c ) ^c ) = A

For sets A , B and C if the sets are finite :-

• n ( A U B U C ) = n ( A ) + n ( B ) + n ( C ) - n ( A ∩ B ) - n ( B ∩ C ) - n ( A ∩ C ) + n ( A ∩ B ∩ C ) .

De Morgan's Law for two sets :-

• ( A U B ) ^c = A^c ∩ B^c
• ( A ∩ B ) ^c = A^c U B^c .

Note :- The sign of complement for a set A is written in three ways like A^c , A^Bar and A^dash . I am not able to write sign for Bar and dash . Dash symbol is like that of negative ( X - axis ) .

I hope you will read my full answer .

Answer 2

Answer:

the greatest value of n ( A U B ) =40

the least value of n(A ∩ B) =5.