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1. Set X:= {1,2,3). For each of the eight subsets U of X, define (U) to be the set of all permutations of X satisfying u" = u for each element u in U. For each of the six subgroups H of Sym(X), define (H) to be the set of all elements x in X satisfying x1 * = r for each element in H. Show that (0,7) is a Galois pair with respect to the set theoretic containment on the power set of X and the set theoretic containment on the power set of Sym(X). Determine the galois elements with respect to (7,0). 1. Preserving and reversing orders Let X be a set. A binary relation on X is called an order if it is reflexive, antisymmetric, and transitive. If r is an order on X, the set X is called ordered with respect to r. If it is understood with respect to which order the set X is ordered, we just say that X is an ordered set. If X is ordered and a pair (y, 2) of elements of X belongs to the order with respect to which X is ordered, we write y