1. Assuming that xy=x+y+1 and xy = xy +x+y are two binary operations defined for all x, y ER, i) prove that both and are commutative; ii) prove that both and are associative; iii) prove that is distributive with respect to ; iv) verify if and have neutral elements; v) decide which numbers x E R have their inverse elements with respect to and , and which do not. 2. For what values of parameters a, b ER the operation xy = xy+ax+by+2 defined for x, y E R is commutative and associative? 3. For G = {1, 3, 5, 9} and xoy defined as the last decimal digit of the number xy+3x+3y - 4 i) show that o is a well-defined operation on G and present its table; ii) find the neutral element of o; iii) prove that (G, o) is a group. 4. For H = {1, 3, 7,9} and xo y defined as the remainder left over when the number xy + 4x + 4y is divided by 12 i) show that o is a well-defined operation on H and present its table; ii) find the neutral element of o; iii) prove that (H, o) is a group. 5. Let P be the polygon with vertices (1, 0), (2, 0), (4, 1), (4, 2), (3, 4), (2, 4), (0,3), (0, 2) taken in that order. Find all isometries of P. What group do they form? 6. Let Q be the polygon with vertices (1, 0), (2, 0), (4, 2), (4, 3), (3, 4), (2, 4), (0, 2), (0, 1) taken in that order. Find all isometries of Q. What group do they form? 7. Which of the groups introduced above in Problems 3-6 are pairwise iso- morphic and which are not? 1 table as in the multiplication table, not the piece of furniture