Problem

For a coplanar set \(S\) whose points satisfy \(XP=r\) for fixed \(P\) and \(r>0\), state the precise circle-membership conclusion and subset notation, and distinguish lying on the circle from being the entire circle.

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condition = XP=r with r>0; conclusion = P lies on circle; subset notation = P∈S; completeness caveat = center is radius r away

condition = XP=r for every X∈S; conclusion = S is automatically the whole circle; subset notation = S equals the circle; completeness caveat = two points suffice

condition = XP=r for every X∈S; conclusion = every point of S lies on the circle centered at P with radius r; subset notation = S⊆X in the plane | XP=r; completeness caveat = S is the entire circle only if it contains every coplanar point at distance r

condition = XP=r; conclusion = points lie in disk; subset notation = XP≤r; completeness caveat = boundary and disk are same

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For a coplanar set \(S\) whose points satisfy \(XP=r\) for fixed \(P\) and \(r>0\), state the precise circle-membership conclusion and subset notation, and distinguish lying on the circle from being the entire circle.

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