If weight (H ) < weight (H j ℓ ), then by Lemma 0.1, H ∪ H′ will be a t-cover hitting set with a smaller weight than the weight of H ℓ, which causes contradiction.
Based on this analysis, it was suggested that biases of data sets might cause contradictions in studying mechanisms.
Most of the possible settings would cause contradictions and be discarded, leaving only a few to be investigated in detail.
Simultaneous occurrence of these two activities can cause contradictions.
Similarly, it will cause the contradiction as before.
This causes a contradiction to the definition of "Pareto-optimality".
For any initial value for r 1, if r 2=r 1, Algorithm 3 converges and the rank detection result is r 1, which causes a contradiction.
Since in our assumptions, (X_{ij}) and (Y_{ij}) are mutually independent, the assumption of independence between (delta_{i,i-1,j}^{(1)}) and (delta _{i,i-1,j}^{(2)}) cannot cause a contradiction.
Furthermore, as any S ∈ T X [ 1 : i ] for i < i q, h i t(H′)[ S] = t (otherwise, it would cause a contradiction that H ℓ is a solution as no element in { u i q, …, u i ℓ } will cover S).
Note that an incorrect classification into the Hgs might cause a contradiction to the assumption of independent occurrences of mutations, but this is not a real concern, since Hg classification is done with very high confidence according to SNPs (Hammer et al., 2009; Karafet et al., 2008; The Y Chromosome Consortium, 2002).
