18 Prove or disprove Let ABC and D be four sets and A C B

18 Prove or disprove: Let A,B,C and D be four sets, and |A| = |C|, |B| > |D|, then |A - B|

Solution

|A| =|C|

This implies no of elements in A = no of elements in B. Let this be equal to n

As |B|>|D| , there is atleast one more number of elements in B than D

If b is n(B) and D is n(D) then b>d

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A-B can have no of elements as either n-b or n

i.e. n-b<|A-B|<n

Similarly

n-d<|C_D|<n

As b>d, n-b <n-d

Hence it follows that

|A-B|<|C-D|