1) Let R and S be rings. Show that the cartesian product of sets Rx S = {(r,s) |r € R, s E s} with the addition (r, 8) + (r', s') := (r + r', s + s') and the multiplication (r, s) · (r', s') := (r · r', s · s') is a ring.

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A: Here we prove a equivalent statement.

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Q: (b). Let Y = {a, e, o, i, u}, take two relations R and S on Y and Prove that (SoR)-1 : R-1os-1.

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Q: True or False? Let R be a ring and a, be R. Then a^2 - b^2 = (a-b (a+b) v a,b € R

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A: Given vectors are Here observed that So we have each of .In other words ,