Suppose that we have three variables X₁, X₂ and X₃, which are defined on the same domain of {1,2,3). Two binary constraints for these three variables are defined as follows:
R₁₂ = {(X₁, X₂), (2,1),(2,3), (3,2), (3,3)}]}
R₁₃ = {(X₁, X₃), (1, 2), (2, 1), (3,1),(3,3)]}
1. is X₁ arc-consistent with respect to X₂? And is X₁ arc-consistent with respect to X₃? and why? In addition, is X₃ arc-consistent with respect X₁ if the constraints between R₁ and R₃ are undirected (i.e., R₃₁ is defined as {(X₃, X1), (2,1),(1,2), (1,3), (3, 3)]} that switches the element order of every two-tuple of R₁₃)? and why?
