r(A)>=ns(A), no of values obtained by adding/subtractin any 2 elements

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r(A)>=ns(A), no of values obtained by adding/subtractin any 2 elements

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#1 h Mar 1, 2020, 12:41 AM • 1 Y

Y by Mango247

For a finite set $A$ of integers, define $s(A)$ as the number of values obtained by adding any two elements of $A$ , not necessarily different. Analogously, define $r (A)$ as the number of values obtained by subtracting any two elements of $A$ , not necessarily different.
For example, if $A = \{3,1,-1\}$
$\bullet$ The values obtained by adding any two elements of $A$ are $\{6,4,2,0,-2\}$ and so $s (A) = 5$ .
$\bullet$ The values obtained by subtracting any two elements of $A$ are $\{4,2,0,-2,-4\}$ and as $r (A) = 5$ .
Prove that for each positive integer $n$ there is a finite set $A$ of integers such that $r (A) \ge n s (A)$ .

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