For a set in a metric space and 0"> 0 , denote by ( ) the set of elements that are -far from . In property testing, a -tester for is required to accept inputs from and reject inputs from ( ) . A natural dual problem is the problem of -testing the set of ``no'' instances, that is ( ) : A -tester for ( ) needs to accept inputs from ( ) and reject inputs that are -far from ( ) , that is reject inputs from ( ( )) . When = ( ( )) the two problems are essentially equivalent, but this equality does not hold in general.

In this work we study sets of the form ( ( )) , and apply this study to investigate dual problems in property testing. In particular, we present conditions on a metric space, on , and on a set that are sufficient and/or necessary in order for the equality = ( ( )) to hold. Using these conditions, we derive bounds on the query complexity of several classes of natural dual problems in property testing. These include the dual problems of testing codes with constant relative distance, testing monotone functions, testing whether a distribution is identical to a known distribution, and testing several graphs properties in the dense graph model. In some cases, our results are obtained by showing that = ( ( )) ; in other cases, the results follow by showing that inputs in ( ( )) are sufficiently close to . We also show that testing any dual problem with one-sided error is either trivial or requires a linear number of queries.