Here I am assuming that the sets A1,A2,A3,......,Am are all distinct
THE QUESTION IS WRONG
The number of distinct sets of the form Ai ⊕ Aj should be at least m-1.
Take the example
A1 = {1,2,3} A2 ={3} A3 = {1} and A4 = {2}
TRY TO FIND 4 distinct sets of the form Ai ⊕ Aj. YOU CANT FIND IT THERE IS ONLY 3.
So, I am proving that.
Now, let us assume for arbitrary i,j ∈ { 1,2,3,.....,m}
we have A1 ⊕ Ai = A1 ⊕ Aj
=> A1 ⊕ (A1 ⊕ Ai) = A1 ⊕ (A1 ⊕ Aj)
=> (A1 ⊕ A1) ⊕ Ai = ( A1 ⊕ A1 ) ⊕ Aj ( Assosiative property XOR)
=> Ai = Aj
But we know that all the Ai 's are distinct.
So, our assumption was wrong
Thus (A1 ⊕ Ai ) != (A1 ⊕ Aj)
Thus (A1 ⊕ A2) , (A1 ⊕ A3) , (A1 ⊕ A4) , ..................., (A1 ⊕ Am) are all distinct sets.
So, we have got m-1 distinct sets of the form Ai ⊕ Aj
Thus there are at least m-1 distinct sets of the form Ai ⊕ Aj (proved)