A nonempty set
of real numbers is called a
-set if the conditions
and
imply that the sequences
and
are identical up to a permutation. Let
be infinite sequences of real numbers with
where, for a set
of real numbers,
denotes the difference set
Prove that if
is a
-set, then
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A nonempty set $A$ of real numbers is called a $B_3$-set if the conditions $a_1, a_2, a_3, a_4, a_5, a_6 \in A$ and $a_1 + a_2 + a_3 = a_4 + a_5 + a_6$ imply that the sequences $(a_1, a_2, a_3)$ and $(a_4, a_5, a_6)$ are identical up to a permutation. Let
$$A = \{a(0) = 0 < a(1) < a(2) < \ldots \}, B = \{b(0) = 0 < b(1) < b(2) < \ldots \}$$
be infinite sequences of real numbers with $D(A) = D(B),$ where, for a set $X$ of real numbers, $D(X)$ denotes the difference set $\{|x-y| | x, y \in X \}.$ Prove that if $A$ is a $B_3$-set, then $A = B.$
Školjka