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Binary operations: Inverse
Binary operations: Inverse
Last updated at Jan. 28, 2020 by Teachoo
Misc 14 Define a binary operation *on the set {0, 1, 2, 3, 4, 5} as a * b = {β(π+π, ππ π+π<6@&π+π β6, ππ π+πβ₯6)β€ Show that zero is the identity for this operation and each element a β 0 of the set is invertible with 6 β a being the inverse of a. e is the identity of * if a * e = e * a = a Checking if zero is identity for this operation If a + b < 6 Putting b = 0 a < 6 This is possible Now, a * 0 = a + 0 = a 0 * a = 0 + a = a Thus, a * 0 = 0 * a = a So, 0 is identity of * If a + b β₯ 6 Putting b = 0 a β₯ 6 This is not possible as value of a can be {0, 1, ,2, 3, 4, 5} Now, we need to show that each element a β 0 of the set is invertible with 6 β a being the inverse of a. a * b = {β(π+π, ππ π+π<6@&π+π β6, ππ π+πβ₯6)β€ An element a in set is invertible if, there is an element in set such that , a * b = e = b * a Putting b = 6 β a So, a + b = a + (6 β a) = 6 Since a + b β₯ 6 a * b = a + b β 6 a * b = a * (6 β a) = a + (6 β a) β 6 = 0 b * a = (6 β a) * a = (6 β a) + a β 6 = 0 Since a * (6 β a) = (6 β a) * a = 0 Hence, each element a of the set is invertible with 6 β a being the inverse of a. s