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- what are the congruence classes in the set of intergers?Suppose (X,≺) is a well-ordering. Then (X,≺) is not isomorphic to (pred(X,≺,x),≺) for any x ∈ X.A partition P of Z is said to be compatible with + if given any X, Y ∈ Pand any x1, x2 ∈ X, y1, y2 ∈ Y the elements x1 + y1 and x2 + y2 belong to the same memberof P. Suppose that P is a partition of with 4 members compatible with +. Show that P isthe partition associated to the equivalence relation congruence modulo 4. Can you see how togeneralize this to a partition of Z compatible with + having an arbitrary amount of classes(not just 4).
- Suppose that Z10 and Z15 are both homomorphic images of a finitegroup G. What can be said about |G|? GeneralizeLet X = {a, b, c} and R={(c, b),(a, c)}. Then Check the all properties reflexive,symmetric, antisymmetric, transitive etc. Which one is true?Suppose that A and B are subsets of S. Express symmetric difference of A and B (AtriangleB) in terms of intersections, unions, and complements.