Binary Operation Worksheet and Solution

Below you could see some problems based on binary operations. Solved Examples Question 1: The binary operation * defined on Z by x * y = 1-2xy. Show that * is cumulative and associative. Solution: Given x * y = 1-2xy Binary operation is cumulative, since x * y = 1-2xy = 1-2yx = y * x => x * y = y * x * is cumulative. Now, check * is associative x * (y * z) = x * (1-2yz) = 1-2x(1-2yz) = 1-2x + 4xyz and (x * y) * z = (1-2xy) * z = 1-2(1-2xy)z = 1-2z + 4xyz => x * (y * z) ≠≠ (x * y) * z Thus, we can find that * is not associative on Z. Question 2: The binary operation * defined on Z by x * y = 1 + x + y. Show that * is cumulative and associative. Solution: Given x * y = 1 + x + y Binary operation is cumulative, since x * y = 1 + x + y = 1 + y + x = y * x => x * y = y * x Therefore, * is cumulative. Now, check * is associative x * (y * z) = x * (1 + y + z) = 1 + x + 1 + y + z = 2 + x + y + z (x * y) * z = (1 + x + y) * z = 1 + 1 + x + y + z = 2 + x + y + z x * (y * z) = (x * y) * z Thus, * is also satisfies associative property. A binary operation on a set is a calculation involving two elements of the set to produce another