I recently read this article about a property of powers of 2 which was the work of Erdos and Selfridge.
They define the problem using the concept of multisets and the statement is referred to about two multisets A and B where elements can possibly repeat. I am trying to figure out the answer to the following question.
For a set A, define the set A2 to contain the pairwise sum of distinct elements of the set A. Note that here, both A and A2 are sets and not multisets. Hence all the elements are guaranteed to be distinct. If it is given that A2=B2 for two sets A and B with finite number of elements containing positive integers, is it necessary that cardinality of A and B be equal?
I am finding it hard to answer the above (seemingly easy) question. I could only observe that if elements of A and B are arranged in increasing order, then the first two and last two elements of A and B must sum to the same value.
Please let me know as to what the answer to the above question is and also a proof or a hint to the proof would be really helpful.
If you write down a long enough sequence of integers, you'll find that the middle elements of $A_2$ can be built in many ways, and the middle elements of $A$ only generate the middle elements of $A_2$, so some of them can be dropped. For example
$A = \{1,2,3,4,5,6,7,8\}, B = A \setminus \{5\}$
both generate $A_2 = B_2 = \{3 \dots 15\}$.